My MagSafe adapter has suffered from strain relief and the insulation is completely frayed off near the tip (it is the newer L-shaped adapter rather than the old T-shaped adapter). It still seems to charge the computer. Is there any risk to my physical safety from continuing to use it?
Frayed jacketing represents a potential fire hazard if the wires themselves become exposed and overheat, short, and/or arc. Provided only the jacketing is frayed, you could use electrical tape to provide protection to the internal wires or you could add a sleeve or heat shrink tubing.
A safe answer is to order a replacement from Apple in order to protect yourself, your device, and your property.
The same voltages that you find on those wires will also be present on the contacts of the magsafe connector itself. Since these are accessible to a finger, you can be pretty sure that there are no harmful voltages there. I think the biggest potential danger is if the wires short out and enough heat is generated to start a fire. Even that possibility sounds rather far fetched to me, but I won't totally rule it out. The power adapter ought to contain circuitry to protect against that, however.
(I don't usually add disclaimers to my scribblings on the net, but one seems in order here: I am not an expert, just a moderately competent amateur in the field of electronics.)