Forget about cheap and close to the center. Unsafe areas are considered Zuidoost, Bijlmer, Diemen, etc. My favorite neighborhoods are OudZuid, Jordaan, OudWest, and basically everything next to the center.
There's also Amsterdam Noord, which is cheaper, and it looks far, but having the ferry 24h, you can get to central station faster than from others.
Just use Google Maps to calculate how much time it takes to your work by bike (which is the best transport here). Under 20 mins is perfect.