Well what do you want to spend per night on a budget hotel? Not everyone has the same idea of what "budget" means. What is your budget price per night?
There are hundreds of hotels in Bangkok in all price ranges. What area do you want this hotel in? Are you male or female? Some good hotels are near nightlife - for some people this is great and for other not so great! Some places young travelers will stay in because of cost older tourists would not want to be there!
My advice is if you are really young - teenagers or close to it - choose the Khao San Road area where you will have many other kids close to your age. Slightly older or more mature maybe a hotel in the Siam, Sukhumvit Road or even the Pratunam area where you can walk to malls, street vendors, restaurants, etc.
If you have a hotel not far from the Skytrain route you can use that train service to get around the city. No Skytrain in the Khao San Road area.
Airport Rail Link, subway and Skytrain route.
http://www.urbanrail.net/as/bang/bangkok.htm
Here are a few hotels you can consider.
http://www.sawasdee-hotels.com/index.html
http://www.staywithorchid.com/
http://www.udeebangkok.com/
http://www.thaiwebsites.com/budgethotelsbangkok.asp
http://silom.lubd.com/
Many more suggestions on this link:
http://www.tripadvisor.com/SearchForums?scope=2&ff=3686&geo=293915&q=cheap+hotel+bangkok&sub-search.x=0&sub-search.y=0&sub-search=Click+to+search+forums&returnTo=__2F__
Good luck.