Do these specks wash off?

They do seem to wash off with a mild scrub... they don't just rinse away. The damage to the leaf remains.

On basil, could be downy mildew.

Please post another image (the top leaf surface) and of an entire plant with the problem

Edit:
Also please post images of the other plants as it may be several different things.

This post was edited by jean001a on Sat, Jun 22, 13 at 14:18

Here is an overall picture of the basil. Damage to the upper leaf surface can be seen. I had a couple basil plants die off a few months ago with similar symptoms, so I'd like to avoid losing this one.

The morning glories seem to be particularly susceptible to this as well. Here's a good indication of the damage:

Possible on basil,but I'm not certain.

Review this info from Cornell to see if it's a match
http://vegetablemdonline.ppath.cornell.edu/NewsArticles/BasilDowny.html

The morning glory looks different. Perhaps environmental.

I'm curious; are you growing the plants indoors or outdoors?

The basil is outside, but the morning glories are indoors. The grapes and mint are also outside. I had assumed that it was the same thing attacking all these, but that might not be the case.