I see ants crawling into the eye, should I stop them? Can I even stop them? I hope they don't do any damage to the fruits?

Yes you should stop them. If you can. The ants themselves are a small part of the problem. The organisms on the ants like yeast will speed up the souring process in the fig. Yeast and the sugar in the fig do not go together well. You can search on this site about Tanglefoot and its usage. It will help prevent the ants from getting up the trunk. A search using the keyword ants should also show other information. 

Thank you so much, nauvillus.  I'm glad I asked because I was still thinking if it might be a silly question since ants are very much a 'natural' thing in the garden.  Did a search like you said and, my, I have learned so much.  Thanks.