Why demand Noahide Courts to Enforce Capital Punishment When The Sanhedrin Has Chosen To Not Execute capital Punishment?
Good question. Some background first – The Torah does prescribe capital punishment as the penalty for transgressing the Noahide laws. This is only applied to violations of the explicit model 7 commandments, and is rarely required for the derivative laws learned out from the seven Noahide laws. All in all, there are only about 20 specific crimes from which a Noahide would be subject to the Death penalty. However, Jews are in a MUCH stricter situation. They have far more prohibitions for which they may be executed or face corporal punishment than Noahides.
As you correctly observe (below) the death penalty was rarely carried out for these transgressions, though. However, the failure to carry it out was not the result of a choice made by the Sanhedrin or any other court. Rather, in addition to the required penalties, there are a number of laws regarding evidence, the burdens of proof required to put someone to death, the means of verifying the evidence, etc. Unless all of these procedural and substantive requirements are met, someone could not be executed. It was exceedingly rare to have a capital case in which the actual requirements and burdens of proof were met. Therefore the death penalty was rarely carried out in practice. The Sanhedrin was not choosing to abandon the death penalty. Rather, there were many requirements that had to be satisfied for it to carried out.
The situation is not that different for Noahide courts either. Just as the Sanhedrin, Noahide courts must carry out the death penalty. Also, like the Sanhedrin, there are rules of evidence and burdens of proof that have to be met in order to carry it out. While these burdens are different that those in a Jewish court, they have much the same effect: it would be very rare for a Noahide court to every actually carry out the death penalty.
Should Noahide Courts Be more Stringent In Carrying Out The Letter Of The Law?
To continue the above explanation – the issue isn’t that the Sanhedrin is being lax. Rather, the death penalty is only carried out when it is clearly sustained under the law. This is a high burden for both Jewish and Noahide courts.
Rabbinic attitudes concerning the death penalty are also reflected in statements such as "a Sanhedrin that effects an execution once in seven years is branded a destructive tribunal." Rabbi Elizer Ben Azariah said "once in seventy years." Rabbis Tarfon and Akiba said, "If we were members of a Sanhedrin, nobody would ever be put to death." In that same Gemarra, however, Rabbi Simeon Ben Gamaliel dissented: "If we never condemned anyone to death, we might be considered guilty of promoting violence and bloodshed.... [We] could also multiply shedders of blood in Israel" (all Makkot 7A).
Rabbis Tarfon and Akiva held that the procedural and evidentiary burdens required were so high as to be virtually impossible to meet and they intended to always err in favor of the defendant in their own judicial discretion. Noahide judges would have the same discretion when dealing with ambiguities.
Forty years before the fall of Jerusalem in 70 C.E., the rabbis abolished capital punishment altogether (Soncino Talmud, Sanhedrin page 161, footnote 10). Rather than applying the four methods of execution themselves, they ruled that punishment should be carried out by divine agencies (Sanhedrin 37B, Ketubot 30A, & 30B). In other words, a punishment so awesome as the taking of a person's life should not be entrusted to fallible human beings, but only to God.
By not carrying out capital punishment are we the ignoring the Laws of Torah?
A major problem after the fall of Jerusalem and loss of the temple is that the Sanhedrin and court system no longer had jurisdiction over many matters. Since their jurisdiction was in doubt, they ceased to administer the death penalty. This was a matter of law, not of their own discretion.
More on Noahide Dinim From the Noahide Laws Yeshiva Course
The Basics of Dinim
The mitzvah of dinim, civil law, is one of the trickiest of the Noahide laws to both define and understand in terms of its real world applications. Much of this difficulty is historical in origin. Since the Jewish world has always maintained and used its own religious courts to judge monetary disputes, there was never a practical occasion or need to address the Noahide laws of dinim. This was the case until 1550 when a legal dispute prompted a massive evaluation by scholars of Noahide dinim.
Although the Talmud reads the earliest reference to dinim from Genesis 2:16, the Torah is abound with references to the concept and need for justice. For example, Genesis 9:5-6:
I will certainly demand the blood of your lives; at the hand of every beast I shall require it, and at the hand of man, even at the hand of every man's brother, I shall require the life of man. Whoever spills a man's blood, by man shall his blood be spilled...
This verse clearly states a judgment and punishment for a murderer, requiring the punishment to be carried out at the hands of man. The Midrash expounds upon many other examples of pre-Sinaitic expectations for justice. Maimonides distills these allusions into the following description from Hilchos Melachim 9:14:
How do the gentiles fulfill the commandment to establish laws and courts? They are obligated to set up judges and magistrates in every major city to render judgment concerning these six mitzvot and to admonish the people regarding their observance.
A gentile who transgresses these seven commands shall be executed by decapitation. For this reason, all the inhabitants of Shechem were obligated to die. Shechem kidnapped. They observed and were aware of his deeds, but did not judge him.
Maimonides’s makes three very important points:
1) They are obligated to appoint judges and magistrates in every major city… Dinim obligates Noahides in the establishment of courts. The purpose of these courts, and indeed the essence of dinim, is to establish order between man and his fellow. This is because God places more emphasis on harmony between men than between Himself and man. Rashi points out that this is the reason the generation of the flood and Sodom and Gomorrah were destroyed, while the generation of the Tower of Bavel was only dispersed. In the times of the flood and of Sodom and Gomorrah, the main sins were between man and his fellow. Therefore, they were destroyed. However, in the times of the tower, their sins were primarily between man and God, therefore God was lenient with them.
2) …to render judgment concerning the other six mitzvos… What is the content of the laws of dinim? Maimonides states that these laws are fundamentally procedural: they apply to the courts and consist of rules and methods for administering judgment for the other Noahide laws. It does not appear, according to Maimonides, that dinim includes matters of substantive law – actual prohibitions or demands on societal or individual behavior.
3) … and to admonish the people in their observance. It is a requirement of the courts to engage in public education of the Noahide laws.
According to Maimonides, it appears that Noahide courts fulfilling these three fundamental purposes meet the standards for dinim. However, this proposition is not so simple. The question of content, point #2 above gets us into complicated waters.
See Sanhedrin 56b.
 Gen. 11:9.
 See Chemdas Yisrael 9:29; Machaneh Chaim II:22.
For a far more in-depth and advanced study of all the Seven Noahide Laws consider taking the Noahide Laws Yeshiva Course.