Grip strength is very important for a large number of reasons, so first of all, kudos for identifying the problem.
One of the reasons, as you point out, is that poor grip strength will limit the number of reps you can do, which short-changes your primary mover for that exercise.
The most common mistake is to start using straps in order to take the grip out of the equation. The problem with this is that you're hiding the problem instead of fixing it. In fact, lat pulldowns are a great grip strength builder, so now you've taken away an opportunity to improve on it!
If, however, you start working on your grip with grip-specific exercises, THEN it's fine to use straps until your grip catches up.
How to train grip strength
There are lots of cool and varied exercises you can do to improve your grip. The first thing I would recommend is looking at, is a video by David Horne where he outlines a very good starter routine. It only lasts for a few minutes, and can easily be added to the end of your session.
Also the best part about this routine, is that it's very easy to add more resistance as your strength improves.
I could list plenty of resources, but as it happens, such a list already exists on the Grip Training subreddit.
These guys take grip strength very seriously, and have outlined various routines for anything from beginners to advanced lifters, and also depending on what your motivation is, which in this case is lifting weights.
Here is the "Master List" of routines, and here is the one recommended specifically for those looking to improve grip for weightlifting. It mentions deadlift specifically, but a bar is a bar.