Having lived in Dubai, I can tell you that it is a very safe place in general with a fair amount of police presence so you would be fine in most areas.
Having said that, and keeping in mind that you have a decent budget for how much you can spend at a hotel/apartment every night, I would encourage you to stay at a hotel. The reason I say this is because you won't have to worry about room cleaning, etc. as any decent hotel will prop things up while you are sight-seeing during the day. Additionally, it would be more holiday like as opposed to apartment rentals because you'd be exposed to all kinds of tours and even other tourists.
Now, for some actual recommendations. I've always seeing Dubai as a place that has "2 downtowns".
The old downtown (known as "Bur Dubai") is where Dubai actually first sprung up in the intial years and is bustling with shops and people all within walking distance. If you like the sound of this, I recommend the following affordable hotels or apartments:
Both of the above are very close to markets and a lot of restaurants.
The new downtown is where a lot of the well known structures such as Burj Dubai have come up. If you choose to live around this area, you wouldn't walk around as much. You are more likely to be taking a cab around. In this case, I recommend the Ramada Chelsea Al Barsha Hotel.
Few other things to keep in mind:
- Dubai is a relatively small city. To get from the "old downtown" to "new downtown" could be as less as a 15 minute car ride.
- The U.A.E. as a whole is a small country. You can even get to the capital Abu Dhabi is less than 2 hours.
- There are things to do and see in most parts of Dubai. Whether it is the busy markets or the extravagant shopping malls, it is all there and just a taxi ride away.
- Dubai is a relatively "new" city which means most hotels are relatively new and clean (even the not-so-expensive one's). However, if you play it right, you can stay at a 4 or even 5-star hotel for a lot less than $180 a night.
Hope this helps!