At Bamburi beach there is a Barclays Bank with 2 ATMs between the Travellers and White Sands Hotels. 

 If these are not working there are ATMs from 5 or 6 banks at the City Mall (Nakumatt complex), a short taxi or matatu ride away.

 If you want to exchange currency the best rate is at the Shaban Mini-Market, a mini-supermarket in the row of shops between the filling station and Cheers Bar. You can also change money in the Cheers / Unik office right next to Cheers bar usually also a good rate. You can, obviously change money at the banks though the rate will not be as good.

 You can reach Shaban from the beach.  There is a narrow path between Sai Rock Hotel and the little hut that rents out rubber inner-tubes, swim suits and plastic chairs. The path comes out at the filling station. Between the filling station and the main road the parade of shops including Mombasa North Coast Tours, a pharmacy, a betting shop, and Shaban Mini-Market on your right