There are a couple of hostels there I'm aware of - St Christopher's Inn and the Dover Castle. I've never stayed in either but have had drinks at both on nights out and both seem like they'd be pretty cool places to stay. St Christopher's Inn in particular pretty lively and right on Borough High Street. Borough itself is a great area with the market, good nightlife and some great old London pubs in the market area. Bermondsey Street a short walk away and also very cool, as has been mentioned. It's also very easy to explore other parts of London from Borough. You can either jump on the Northern Line with easy access across the tube network or walk up through the market to the Southbank (of the River Thames), wander over the Millennium footbridge towards St Paul's etc.