I've not seen inside many hotels but none of those I've seen had mosquito nets - though they were air conditioned rooms. If you have an air conditioned room, make sure you keep it cool/cold enough to keep the mosquitoes out. We didn't keep the room particularly cool this trip and got the odd bite in the room.
If you can, take a knock down fly spray with you and give the room a good dose on arrival. if in doubt spray it each evening, as twilight is when the majority of mosquitoes become active. Fly sprays are readily available in Goa but if you take one with you it's ready for immediate use without having to go shopping.
The only beach hut I've used had a mosquito net though it was poorly designed and I didn't have a lot of faith in it.
As for insect repellents - the locally produced Odomoz seems quite effective and is cheap and readily available. I used that or a 50% Deet spray on exposed skin in the evenings - I didn't bother during the day if we were in the coastal areas but would if travelling inland, especially in the south of the state where day flying mosquitoes are more prevalent. I sprayed my clothing with Deet as well the day I went horse riding.
The recommended practice is to spray your clothing with 100% deet to avoid being bitten through it. I don't normally bother but on our last night in Goa I got several bites on my back (through a shirt) whilst dining out, so perhaps I should follow the advice in future!
Take sensible precautions to avoid being bitten and you should be fine. There is malaria in Goa but it's far from endemic and historically has been associated with the migrant workers who lived and worked in poor conditions. Many visitors take no precautions at all and come to no harm, but that's not a course of action that can be recommended.