My goal is to draw a semi-transparent curve. User moves cursor and I draw the curve under the cursor.
I've tried to use antialiased points to draw line, but I don't know how to make it transparent. I can't use lines to draw the curve, because can't set both antialiasing and line width.
Should I use triangle strip to draw curve?
Yeah, if you want to do a nice job with this, you could tessellate your wide curve into a triangle strip. There are many papers written about stroke tessellation.
You can then texture your triangle strip with a square alpha texture that has a nice solid, anti-aliased circle in it -- this causes the wide line to appear anti-aliased! Check it out:
http://homepage.mac.com/arekkusu/bugs/invariance/TexAA.html
Very cool stuff.