There is no trouble in proving a negative that does not exist for proving a positive. For example, you can prove to your own satisfaction that there are no monkeys on your shoulder, even though that is a negative claim. The problem of proof to which you alluded with your example is just the famous problem of induction, which exists for both positive and negative claims.

The negative claim that so-and-so has never eaten a baby is difficult to prove because that claim is actually a group of claims: so-and-so was not eating babies at time one, time two, time three, etc. Each and every claim in that sequence could be proven with a photograph that shows so-and-so doing something other than eating babies. Each negative in that sequence is provable. The problem is the length of the sequence: you need hundreds and hundreds of proofs. You would have the same trouble with any similar sequence of positive claims, such as the sequence entailed by the statement that there is a human-habitable planet in one out of ten solar systems.