In theory, gate-checking is less wear-and-tear on the stroller than desk-checking. Because a stroller is not a safety-device, as is a car seat, do whatever is more convenient for you. You can bag the stroller, throw on a few "fragile" stickers (who knows what good it does) or whatever else you want to do. The cover is a good idea if it's raining. Double secure it with a bungee cord to keep it closed en route. It's car seats which are a bad idea to desk-check so keep that with you and use it on board.

If you stroller has foam handles, you may want to protect those, perhaps with some bubble wrap or just plastic and duct tape. Mine ripped once and this is common. You might even want to remove them. Also, I used to take everything off the stroller I could whether I checked or used the stroller in the airport. Less stuff means less damage.

I used to strap my car seat to a metal luggage cart with an extra bungee and my daughter could ride in it. Then I had a good baby carrier for when she got clingy, to save my back and keep my hands free. Luckily both liked walking and riding better!