I had to rush over to Miami for 4 days in March and was looking for a rather inexpensive hotel to stay in that was reasonably close to the main shopping areas. I eventually decided upon La Quinta Inn (Airport West) due to it's location and average pricing. I was very satisfied with the hotel staff and room at this particular La Quinta and would probably consider staying here again.
Continental breakfast was incuded in the rate (~US$160 /night). There is a free shuttle service to this La Quinta from MIA. This hotel also has free High Speed Internet Access.
There were other hotels in the area but the pricing was not as competitive. Public transportation was difficult since you can't rely on the bus in Miami and it seems that the taxi's don't run that frequent either. You'll probably have to call in a cab 20mins in advance if you're on a strict schedule.
This hotel was about 10mins away from Dolphin Mall as well as Miami International. That would probably run you about US$11 each way by taxi. It might take you about 20-30mins to get to Dadeland mall depending on the traffic. If you've got a GPS or familar with driving in Miami, you'll have no problems getting to and from the malls. There as a mini shopping centre right outside the hotel with a couple dining options however, I did not venture to any of these. A taxi cab ride about 5minutes down the road would lead you to other establishments such as Red Lobster / BBQ Hut and more.
I am still in search for a less costly hotel option. Since I shared this room, it really only cost me 1/2 the price. If I had to go it alone though, I'll probably look for something cheaper.