My bettas have had that same lethargic reaction to a lot of things, so I am going to throw out some suggestions and ask some questions, and hopefully you will be able to better asses what's going on with your fish.
How is your water temperature?
How is your ammonia / nitrite levels?
Are you sure there is nothing physically wrong with the fish? Do the gills look dark around the edges? Does his belly look a bit bloated? Is the end of his tail fat or thin?
Does the water look clear to you?
Is there visible traces of waste or food on the floor?
Can your bettas see each other?
What do you treat your water with?
The best immediate action you could take is a water change. 25% should be okay, but if you have a waste remover, you can do up to 50% without stressing out your fish. Use your judgement, any kind of water change usually stresses them out a little.
Salt is a great idea since it helps kill bacteria in the water and promotes fish health, however if your fish is already having some problems, you may want to start with only a little. Bettas in general have good salt tolerance, but some bettas will react badly to salt regardless, especially if they haven't had it in their water before.
I have only ever used aquarium salt, although I have heard that you can use epsom salts. Don't quote me on this, do your own research or hear from someone more experienced than me about the epsom salt thing. Don't use sea salt or table salt, as it will dehydrate the fish. I personally wouldn't take the chance of using anything other than aquarium salt, but I am very overprotective, and there are probably substitutes that are perfectly fine. Like I said, see if you can hear from someone who knows more than me about salt.
So here are the things my fish have reacted to:
Bad quality water
Disease (Bloating in the belly which can be a sign of dropsy, white spots on face, gills, fins, body, which is a sign of ich)
Malnutrition (Thin tail)
Gill burn (Dark gills. Caused by bad quality water)
Stress (Can be caused by all of the above, as well as exposure to the other bettas.)
MelaFix (Fish medication that is actually dangerous for bettas)
Untreated water (Tap water without a de-chlorinator, water softener, or other treatments to remove harmful elements)
Water with too much salt.
I'm sure I haven't covered everything, but hopefully this will help.