How to unlock the sewers

In order to unlock the sewers in Stardew Valley you need the Rusty Key. You get this in the mail from Gunther once you have donated 60 unique items to the museum, so go ahead and drop of all your artifacts and minerals. Once you have the key, go to the sewer entrance and use it, and it will be unlocked forever.

What can you do in the sewers

There is not a whole lot of content in the sewers, but there are a few things to do. Firstly you can catch the legendary fish Mutant Carp in there. This is one of the five legendary fishes, and can only be caught once in each save file.
Secondly you get access to Krobus, which sells a few items. You can also give him gifts to increase your friendship with him.

Krobus in the sewers.

Full list of items Krobus has for sale

Void Essence - 100g
Solar Essence  - 80g
Slime - 10g
Bat Wing - 30g
Lucky Lunch - 70g
Omni Geode - 300g
Iridium Sprinkler - 10,000g
Stardrop - 20,000g
Bullhead - 200g
Mixed Seeds - 30g
Autumn's Bounty - 400g

Most of the items rotate, so he usually only sells three or four items at the same time. The Stardrop can only be bought once.