What is the best option to extend and have wifi works fast in basement?

If your extender doesn't always get recognized or has connection failures upgrading to a faster internet service will not help at all.

Upgrading the router or extender isn't likely to make a big difference.  Some are better than others of course so it might help.

Three things I'd try:

1.  Move the extender.  They should be closer to halfway in-between the router and the spot you want internet rather than next to where you want it.

2.  More extenders or one of those "Mesh" kits.  You might have to chain extenders together to get decent coverage in the basement.  Needing multiple extenders is common in larger homes.  The mesh kits tend to work a bit better than an ad-hoc collection of extenders since they're an integrated kit and they're designed to work together.  Some of the nicer ones also have dedicated radios for talking to other mesh nodes which can help with performance.  I'd stick to buying equipment that can work with a wired connection in case you end up at #3.

3.  Use a wired connection from the router to an access point in the basement.  It doesn't have to be an actual access point.  Many mesh kits support wiring the nodes to each other.  Most range extenders and WiFi rounters (except for ones that support a cable or DSL connection) have an access point mode.  Please note that Xfinity pods do not support a wired connection back to the router.  Devices marketed as access points are generally commercial or "prosumer" equipment.  They have more features and are more complicated to set up, but are often less expensive than WiFi routers since they don't do as much and don't need as much CPU and memory.

If you don't have existing ethernet cabling to the basement and don't want to run it you can look into MoCA (ethernet over coaxial) adapters and power line adapters.  MoCA can coexist with cable TV & internet.  If you end up using MoCA be sure you have a MoCA PoE (point of entry) filter installed where the cable comes in if you run MoCA on a section of cable that's connected to Comcast.  Otherwise your neighbors might have access to your MoCA devices.  MoCA adapters can go as fast as 2.5Gb.  Powerline adapters don't generally hit gigabit speeds and what you get out of them is highly variable and susceptible to interference since they're sending signals over electrical wiring.