Hi John,
There is a way to select points along a line but it may require a sequence of operations. The "Select by Polygon" tool that usually lives on the "Home" tab has a mode called "Crossing Boundary", and it will work to select anything touching a line, and the "Boundary" object doesn't have to be a closed line. The behavior is that it will select all objects touching the line. You can then filter those objects down to only the points, e.g. using the Selection Explorer or the Advanced Select tool. There is no tolerance option, so it will only select those object directly touching.
If you need to select within a tolerance on either side of a line, I would suggest taking the approach of offsetting the source line left/right by the tolerance amount, using the "Join" command to create a closed boundary from those offsets, and then using "Select by Polygon" with "Inside boundary" selected.
It isn't automatic but it's possible.
