Many cruisers have big dogs.

There are several ways that I have seen to deal with this.

One boat had two dinghies - one was for the dog. Their dog was
trained to climb out of the dinghy using the ladder. Unless the dog
is crippled in some way, dogs can climb ladders.

Some people - especially those that go out for extended periods of
time - train the dog to go on a small patch of artificial turf or
similar, and then rinse it off in the ocean afterwards.

The other possibility of course is to come in to a dock every night,
and rig a ramp to the dock for the dog.

There are doggy PFDs which have a harness handle on the top so you can
grab the dog by the handle to lift him if he falls overboard. That's
better than trying to hold him in your arms - just be sure he can't
slip out of the jacket.