we will be staying in Boppard over night and the next day would like to visit some vineyards for wine tasting as we drive south to Oberwesel, Bacharach and so on. How do I find out which vineyards are open to stop at?
When will you be there? My first suggestion is to drive west to the Mosel valley, the REAL wine country. From Boppard it is only about 10 miles as the crow flies. Lots of wineries. In late summer and early fall there will be "strassenweinfesten" in many villages, with wine makers setting up booths on the streets with wine tastings. There will be strolling bands, many from Holland, in wild costumes. They play at the booths for glasses of wine, then move to the next one.
It is a little different than France or Italy here. The wineries are in the villages in the wine maker's home (basement). The wine maker has a number of plots on the hillsides above the river and commutes to them in his tractor for tending. The harvest is hauled back to the house in the village and the grapes are pressed and the wine is made and bottled in the basement. Each village has its own "brand" and label and all the winemakers in the village use that label. For instance, in the village of Briedel, the local wine is Briedler Herzchen (Little Heart of Briedel).
The wine business works a little different in Germany than it does in, let's say, the Nappa Valley. The grapes are usually grown on public-access land, not a private wine estate. You can usually walk freely among the vines, but there's rarely a winery directly on site (some of the regular evening walks I take with my dog in the warmer months go through such vineyards). The wine maker usually has a workshop somewhere in town, which may be shared among several different producers. Most do not sell directly to the public or operate any kind of showroom, they ship it to a distributor or super market chain. And by volume, German wine is overwhelmingly cheap table wine, sold for about €3-10 per bottle.
That being said, there's a few businesses that offer something like what you're looking for. I don't know the name, but there's castle just downstream of Bingen where you can buy (and I assume, taste) wine. Overall, though, although the Rhine is well-known for wine and tourism, it isn't really a "wine tourism" destination.
As the other poster mentioned, if you travel in late summer, many towns hold some kind of wine festival. The larger ones, like Bad Dürkheim's Wurstmarkt, are comparable to Munich's Oktoberfest, but are usually ignored by the annoying international herd of roving fratboys that infest the Munich celebration every year.
Unlike Napa valley there are few wineries in Europe that allow you to just show up and go on a tour or do a tasting. You can however plan ahead and email producers to set up small tours and tastings.
Schloss Johannisberg has information on their website about setting up a tour and wine tasting. They have a nice tasting room where you can buy the wine directly.
Not far away, Kloster Eberbach doesn’t offer small tours, but has a nice shop that provides tastings. Plus you can tour the very interesting monastery and old wine making halls before or after your tasting.
I’m sure if you research the nicer labels of Rheingau or Mosel Rieslings you’ll find several more wineries worth visiting.