Rashi 42:36 says that Yaakov suspected the brothers of killing or selling Yosef and Shimon. Wouldn't he suspect them doing the same with Binyomin? If so, why didn't Yaakov go with his sons to Egypt to get food knowing the brothers wouldn't harm Binyomin if their father was with them?
There appear to be 2 parts to your question, as I understand it.
First, why didn't Ya'akov go down at all? The simple answer appears to be a combination of items - He was very old. He originally planned to settle permanently in Cana'an - after all that was the land G-d had promised, and it was also the land where Avraham and Yitzhak lived. It's also clear that he planned on dying there and being buried in Ma'arat Hamchpela.
The concern that the brothers would kill Binyamin was, essentially gone via several statements that Yehuda made. Most notable is Breishit 42:9 where Yehuda says that he will be the "guarantor" for Binyamin, and if he doesn't bring Binyamin back and stand him before Ya'akov, then he will have sinned all his (Yehuda's) life. See Rashi's comment on the term "Stand him" stating that Yehuda was stating that "I will not bring him back dead, but rather, alive."
It seems that it was a combination of trusting Yehuda's statement as well as, probably, moreso his timing. Ya'akov was getting more desparate at that point. He and his family were almost out of food! And, Ya'akov realized that sending the children back without Binyamin would probably end up getting all of the children jailed and / or killed.
Ohr HaCha'im commentary states this more strongly. He focuses on the additional word "Gam" - "also" at the end of verse 8 (He says we will live and not die also you, also us..."
Yehuda was stating - It's understood that we can't go back without Binyamin. So, if we don't go at all, all of us will definitely die from starvation. But, if we do go down with Binyamin, there's a possibility that an accident will happen - it's not a definite! And if everything turns out well, at the end, we will bring back food and all of us will return alive, and you will also live from the food brought back. In short, it makes more sense to take a chance and send Binyamin with us.