Is there something we can shot at an atomic bomb to stabilize it?

Once you've got a critical mass of uranium or plutonium, it's going to blow... I don't think something as tenuous as a neutrino beam would help.....

Unlikely.

A fission process involves a nucleus capturing a neutron, becoming unstable, fissioning, and releasing additional neutrons to feed a chain reaction.  That reaction would have to be moderated.  Something that captures the free neutrons, such as cadmium, could help with that, but getting cadmium into an exploding atomic bomb would be problematic.

In a fusion bomb, a fission bomb provides the energy to fuse deuterium and tritium, releasing neutrons, which further augment the fission reaction.  If you inject He3 into the mix, it captures the neutrons and kills the reaction.  Since He3 is a gas, it might be easier to get into a explosion, but it's also unlikely.

I'm not sure what a neutrino beam would do.  First, neutrinos are inordinately unreactive.  Second, they're mostly involved in weak interactions in which quark flavors change.  It seems unlikely that a neutrino beam would have a significant effect on an atomic chain reaction.

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Is this a story idea?  It kind of sounds like one.  If it is, you're better off trying to figure out how to keep the chain reaction from starting in the first place.