So, I need a pair of monitors in basement that will serve both as my occasional studio monitors (mixing my modest synth rig) and also as small PA type speakers for when I do fun music stuff with my kid and run a live mic or iPod through them.
Have a look at the Yamaha HS80m. They are really great for the money and with the 8" membranes you dont have to buy an additional woofer.
I am sure they will fit your needs.
My opinion is go near-field loud speakers but if you were doing more live sets go amplifier. Studio monitors are very critical listening, amplifiers are for live sound. These are fundamentally different and should be treated as such but in your case running an iPod and what not through reference monitors isn’t an issue.
+1 on HS80m’s. I use them for near field monitoring as well as a PA in my small studio. Depending on your room/needs, you may want to add a sub to go with these.