How can one use Thai basil stems and flowers?
First off, I'm afraid you have been using the wrong type of basil for your dish!

In Thai cooking, three different types are commonly used:


"Thai basil" (or "horapha", โหระพา), which tastes a bit like anise / liquorice; it's slightly purple (as shown in the picture in Jolenealaska's answer).
"Holy basil" (or "kaphrao", กะเพรา), which tastes more like pepper/cloves; it can also be a bit purple, and (unlike Thai basil), it is a bit hairy, and the leaves have jagged edges.
"Lemon basil" (or "maenglak", แมงลัก), which (as the name suggests) tastes a bit like lemon.


Notice that "krapow" is just a slightly different spelled transliteration for holy basil ("kaphrao"). Preparing this well-known dish with Thai basil instead of holy basil will certainly make for a very good tasting dish (I've made it myself a few times when I couldn't get holy basil, and in fact I actually prefer it, since I'm quite a sucker for that anise/liquorice taste), but should really be called "Gai Pad Horapha" instead.



As for your question, I typically use the flowers (of either type of basil) to garnish a dish - if I haven't eaten them already while peeling the leaves off the stems.

I typically discard the stems, but on occasion I have chopped them up and mixed them with some honey and ginger (and some mint, whenever I had it), to be used as a dipping sauce for spring rolls.