It depends on the move, but generally your two main options are either to stuff it out (if it has longer start up), or to try to punish what your opponent does next.
For instance Ike's Nair comes out on frame 8. If you manage to get in close on him, you can beat it out with any move faster then that. Sword characters and projectile zoners tend to have close range options that are a bit slower on start up, so they often struggle in boxing scenerios (note that some still have escape options though, like Lucina's Dolphin Slash, so watch out for those).
Similarly, if an opponent wiffs a safe attack you can try to punish what you think they'll do next to protect themselves. If you think they'll shield, you can go for a grab. If you think they'll jump, you can go for an anti-air. This takes some guesswork and also observing your opponent's habits though, so it's not guaranteed. It's important to remember that you won't be albe to punish everything.