Do you have any sharp objects in his tank that he could damage his fins?
How big is his tank?
How much and how often you doing water changes?
Make sure you also rinse/swish the filter in the tank water or dechlorinated water.
babble nest is just natural instinct that doesn't mean he is healthy or sick.
Is he acting healthy? I agree with sainthogan just do extra water changes and see how he is doing. If he is resting more frequently on the bottom then before , if his behavior changed , getting lethargic, hanging on the top or laying on the bottom, refusing to eat do aquarium salt treatment. I didn't see any fin rot on the picture though.
Dark or discolored edges that are ''smudged'' is fin rot. Clear/translucent fins usually -new regrowth.
Give us update so we can decide if you need to treat him with salt.
Aquarium salt you need to pre mix first. Shake it very well make sure it dissolved(not salt crystals left). If you add salt directly to the tank you can burn him.I would use 3tsp /gall with daily 100% water changes. You can treat him in the hospital tank that you have (1 gall) so it easier to change the water.
If your tank is cycled you can do 50% daily water changes in his cycled tank.
Do you know how to acclimate betta to the new water when you do full water change? It very important so you don't shock him with drastic temperature change.