I have an iPhone SE running iOS 12 with the beta profile installed. I don't always update to the next beta when it comes out.
Is there any way to stop updates form downloading as they can take up quite a bit of space?
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I don't believe it is an option without an MDM, but if you use an MDM you should be able to Defer Software Updates for a set number of days.
If you are running macOS Server you can use Apple's Profile Manager, if not I believe there are a few free MDM's if you only have a handful of devices.
Alternatively, you could remove the profile that opts you in to betas. That freezes your device and when you're ready to upgrade, then add the profile back and do the deed, repeating the removal as soon as you restart to prevent the next from arriving.