I bought a couple basil plants last week and planted them out in my garder. A little while after I noticed some sort of mould with white spots on the stem. Does anyone know what this is and what I can do to treat it?

Could be the fuzzy down around the bottom of the plant is possibly a mildew
Basil downy mildew can be transferred by seed, insects, rain or fresh growing leaves. Infected transplants and leaves may not show symptoms if maintained in cool, dry conditions.
Spray with a 50-50 mixture of water/whole milk- 3x's per week. The fatty milk will help control the mildew and pick out the leaves that are effected.

.... could also be an infection by any of the several fungi commonly gathered under the umbrella of damping-off diseases. Sorry for the late reply.
Al