I have wooden blinds throughout in my home (and a little one as well, so I understand the concern for safe cleaners!)
What I do is, I take the blinds outside and squirt them down with the hose, add some Murphy's Oil Soap to an snow brush (I know, sounds strange, but it works like a charm!), rinse clean, and let them dry in the sun. I do this every year. I also use fabric softener dryer sheets in the winter months for a quick clean. The softener on the sheet acts as a barrier, so it actually repels dust.
Hope this helps! Good luck!