Is it possible to blast safes open using dynamite or mines? Or is the only way keys and/or lockpicking?

No, lockpick, keys, or using a terminal are the only ways to get into a locked safe. Also for reference http://www.formspring.me/JESawyer/q/237846437897974391 why.

You either pick the lock, find a key, or hack a terminal to open safes and doors theirs no other way to open them. Just read the above link and I've got to agree making doors or safes breachable would make the lockpick skill useless.

Bummer.