I made a big bet and everyone folds. Following this I mucked my hand and one of the losing players reached into the muck and took my hand to have a look at it. What would be an appropriate remedy or way of handling this situation?
I have seen this happen before, it really depends on the player who mucked their cards. They can either let it slide or call the floor over. It is definitely not cool to look at another players cards if they mucked them (from an etiquette standpoint). I believe dealers are also supposed to protect the muck, I have seen a lot of people try to turn over cards and the dealer prevents them. The rules at different casinos will vary, calling the floor over to make a decision or enforce the rules would be the best option.
The best way to handle this is for the dealer to call the floor over. The floor should explain that no player is allowed to touch another player's cards, and you're especially not allowed to touch the muck, which is in the dealer's possession. If it happens again, the player will be asked to leave. Handling it like this let's the offending player save face, but addresses the problem. The floor isn't addressing him personally, he's addressing everybody. If you're the floor, you should make eye contact with the offending player, but then move on and address the entire table when you make the threat to ban the player. If after this demonstration it happens again, the player should be 86'd for 24 hours. If it happens again after that, he should be barred permanently.
This is a major breach of etiquette. If the player wanted to see your cards, they had the option of paying to do so, and they didn't.
As a dealer, I would have ideally protected the muck proactively, as this is a game security issue generally, though for bigger reasons than maintaining etiquette. That it happened despite (the lack of) this, the supervisor should be called, and they would ideally deal the player out for one round (if by hands) or 15 minutes (if by time).
If the venue does nothing about this, especially after the issue being raised by the affected player, they are showing themselves to be very loose with both security and respect for players, and as a player, I would probably not continue playing there.