First off, note that both words are not used a lot in colloquial. These are rather formal words.
However, in this case, as so often, it's important to understand how the prefixes an- and be- affect the meaning. Admittedly, it's hard to grasp for be-. But basically the prefixes are the opposites to aus- and ent-.
The word pair an-/aus- is pretty simple. It's on/off. You put clothes on or off.
If I had to describe be- and ent-, I would say that it describes a change of state (between opposites), a supply (or withdrawal, respectively), or the idea of (un)covering something.
It depends on the context which description is better. An example would be bewässern/entwässern (irrigate/drain) where you supply or withdraw water to an area of land.
In case of be- and ent- kleiden, you can either think of (un)covering your body or changing from the naked to dressed state or vice versa, i.e. get dressed or undressed.
In the end, it doesn't make a difference if you say ankleiden or bekleiden. Same for auskleiden and entkleiden. But as mentioned in the beginning, they aren't used a lot anyways. So, talk about anziehen and ausziehen.
If you use them, though, be aware that ankleiden is separable whereas bekleiden is not.
Ich kleide mich an.
Ich bekleide mich.
As mentioned in the other answer, there's a second meaning for bekleiden. That meaning is not covered by ankleiden, so you can only go with bekleiden.