RuneScape players in Old School RuneScape have two banking options in Burthorpe. This Old School RuneScape guide expounds upon them.
The first bank is in the guild, and is only accessible with 99 in either attack or strength, or 110 combined attack and strength levels. This bank is on the world map.
However, there is a bank in Burthorpe accessible to all RuneScape players; it is located underground. From the games room, enter the bar immediately south of the steps and descend the trap door into the Rogues' den. There, Benedict Arnold serves as a banker.
The Rogues den is a good place to train cooking, firemaking, and thieving. A permanent fire is located by the bank, there are wall safes, and there is plenty of room to make fires in.
Both banks are still a significant walk from Taverly, although some old school RuneScape players will find Falador closer, depending on what part of Taverly they are at and whether they have the agility to cross the low wall in west Falador.