If you are going to stay in Bogota, keep in mind the location;
Most low-budget visitors choose to stay in La Candelaria, the colonial neighborhood in the center of the city. There are many cheap, nice hostels where you can meet travelers from all around the world. The historic district as well as all the major museums and some nightlife options are within walking distance. The historic hostel is Platypus, which has been open for more than 15 years, but many newer backpacker hostels have opened recently. Ironically, the area is the one of the most dangerous parts of Bogotá and there have been many reports of robberies with knives. Extreme caution is needed when walking at night, especially in front of hostels.
Common precautions apply as in any major south american city: do not walk alone, carry as little cash as possible and leave the passport and credit card at the hotel. Pressure from neighborhood groups to oust the remaining criminals has caused police presence to increase but you must always remain cautious.
Check the location very carefully before you choose a place to stay, security is worse in the tiny deserted streets uphill and closer to Egypto neighboorhood.
You'll find several hotels in the upscale northern districts like Zona T or Parque de la 93. Security won't be an issue but prices are much higher. Nevertheless, you won't have any problem hailing a taxi at 6am in the morning because your hotel would be just around the corner from the nightclub. On the other hand, you can find low to medium price hotels around downtown or near universities (i.e. Chapinero Neighborhood).
Note that in 2008, two foreign travelers were found dead in La Candelaria area due to drugs. Recently, a Colombian broadcasting company CARACOL filmed about drug tourism in La Candelaria, and the program was a big sensation in Colombia.
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