Can I wear my Airborne and Air-Assault badges while in basic training?

Okay, you asking one thing and then saying something else in your follow-up comment. First you ask about wearing your qualification badges and then comment about wearing patches.

I believe there is Reg. that may restrict you from wearing the qualification badges.
Those badges are qualification badges in the same manner as your marksmanship and other badges. As such, if you have not re-qualified, which I believe for Airborne is 5 jumps every year, you should probably avoid wearing them. At least until you can get it details on this question sorted out.
Now, if you are talking about "patches", as in unit patches (i.e., combat patch you have while being assigned to an airborne or air assault unit) then by all means, wear it proudly.
If you served in combat with multiple such units, I believe there are Regs. for which one you are allowed to wear. I think is is the last unit you were deployed with, again get it sorted out before you go off half-cocked and get your ass in some hot water or worse, make a fool of yourself.

My basic had to returning soldiers, in different platoons. One got into trouble for not having rank/badges on uniform, the other talked to cadre and opted only to have rank(lost a rank from E-4 to E-3) stitched to uniform. At graduation, his dress showed badges earned and awards. Some cadre take offense to badges they have not earned, others do not. Better to ask first, at platoon level, and possibly confirm with 1SG out of respect. Either way, those around you will quickly realize you know and understand the program faster than they do, and start asking questions and looking to you for help and guidance.

You decided to come back in for a reason, time for hiding out is over! Wear that shit proudly, you earned them! Airborne over Air Assault, regardless of how many push ups they have you do! Time to start leading from the front!

I would keep a low profile, DI’s have a way of finding out who has had prior service

Unless ordered, you are not required to wear anything beyond name tapes, rank and unit patches. So, keep your badges and medals in your locker, ready for inspection. When they hold your first Class A inspection, wear everything you are authorized to wear and ensure they are documented on your DD214. Returning as an E-4 will be the first tip-off. If you have a Combat patch, that will be the second indicator. By the time you get to your Class A inspection, the cat will be out of the bag.

Although you may believe Basic will be easier, you are an E-4 and will be expected to serve as a Squad Leader, mentor others and assist in any way you can. All of this is good news because when you get your assignment, you may have a Basic training record, indicating service as a Sqd Ldr, ready for E-5. Good luck

Now that you have completed training, how about sharing what choice you made and how it worked for you.

I wouldn't wear them. I had them as a cadet and later a junior officer, having earned them as an enlisted soldier. While they gave me credibility, they also drew a lot of attention to me. Some of it was cool in the beginning. It was nice being respected for what I already knew (I was one of the original NCO Cadre that stood up the 10th Mountain in the early 90's). But that turned into a sore point when less knowledgeable upperclassmen cadets wanted to flex their muscle by trying to trip me up. While they never did, it got old very quickly.

Look, be proud of having those badges and the experience. But keep your head low and just help out your peers. You will get more done for your fellow soldiers if you don't have to worry about getting harassed or worrying about maintaining a higher standard. Put all of that back on right as you get ready to go to graduation ceremonies and kick ass at your next assignment.

No, I wasn’t Army. I started out in the Marine Corps. I earned my Aircrew Wings of Gold. I got out, probably wasn’t my best decision. Hated being a sillyvilian. As the Marines weren’t accepting prior service, I went with the Air Force. No basic training, no tech school, just report for duty at Hill AFB. I’d earned my Wings. The AF said I couldn’t wear them. That only lasted a short time then I said fueck it, I’m wearing them. Never got called on, never got told to remove them.
Wear what you earned. They don’t just hand out these devices. If someone gets a hard one over it, take your hits and move on.

I wore my EIB when I reenlisted. However some higher ranking soldiers challenged me until I got to the 25th ID in Viet Nam. It was harder to get than the CIB.

I was a re-tread as well. It is much better to keep your head down and on the task at hand. There will be plenty of time to strut your stuff after basic!