Well, what do your pathways look like? Are they "below" the grade? Did you dig them out?
Do they wash out?
My paths are mostly filled with 3-4" of gravel and then old cement as stone. I don't care if dirt gets into them as the soil helps everything pack down.
I've never had much problem with weeds in this area since I keep the garden weeded, but when spare seedlings get caught I keep them if they are low growing, and if the are not I either weed them out (I have a cool tool for this) or pour boiling water over them.
A prinkle or spray of a weed suppressant product will help inhibit seed germination.

If you are getting wash out during rains, sloping your walkways 1/2" to the "low" side helps a lot with the run-off.

GGG

another suggestion would be to install that inexpensive black professional edging along the edge of the sidewalk flush with the ground with just the top rounded edge showing. you can get it in rolls at any walmarts or hardware type store. this helps prevent soil runoff from rains etc
to keep the soil off your pathways during planting or weeding i suggest you either cover your walkways with newspaper or a tarp for easier cleanup. most people dont bother though. a good broom sweeping works well if its a concrete or wooden walkway.

It's an old brick pathway that was laid down before I got here..... I've thought of replacing it since I've added a few walkways since I've been here.... I wouldn't care but the dirt drives me nuts with the weeds growing in it.... I have to shovel off at least a few pots or buckets full by the end of the season..... :)

Sounds like it is a little low and the dirt washed in. The black plastic edging might be enough or you can edge with redwood 2x4's if it needs to be higher. That would give a nice finished edge to it.