I use a NETGEAR WNCE2001 WiFi Adapter http://www.netgear.com/home/products/hometheater/networking-for-home-theater-and-gaming/WNCE2001.aspx attached to my BDP-1 which connects it to my WiFi network. Easy one-time setup, works great and costs about $50 US.
Your playlists are saved by the BDP-1 not on your control device. Your iPad, iPod touch, iPhone, etc. will all see the same playlist.
Tony
I can't tell you which is the best and most reliable router. I need a wireless router that works for lots of deviices and has good range.
My experience has been hit and miss. I had a Linksys for about ten years, but then found it did not work with my iPod Touch. I replaced it with a much more expensive Netgear router, which failed in less than two years. I've gone back to a Linksys, an EA4500, which cost about $170. It works better than the other two, but I have no idea whether it will work for years or months.
Because the BDP-1 requires an ethernet connection, all of these routers worked well with it. It seems that if you are using the router only to connect it via an ethernet connection, and then to use your iPod and iPad to control it wirelessly in the same room, or close by, then you probaly would get by with a very low cost one. As to reliability, I have not found any reliable information on which ones are the most reliable. However, if you buy an inexpensive one, then reliability may not be so important.
Dave