Where is the best place to stay in Budapest to get best cultural experience?

That's going to mean different things to different people. All things being equal and if you have no particular driving interest then I suggest you stay within the zone defined here: https://goo.gl/maps/zeG69ZcZSfD2 Its mostly "Budapestian" (is that a world?) with some tourist overtones. At one end you will see Deak Ferenc ter. If you were to calculate the center, based on walking time (or walking and metro time), of all the major points of interest in Budapest it would be close to Deak Ferenc ter.

The significant transportation lines that you will be interested are the M1 metro (the oldest on the continent of Europe), the 47/49 Tram which runs from Deak Ferenc ter to the Great Market Hall and the River. and the 2 Tram which runs the Pest side of the Danube; and to a lesser degree the 4/6 Tram. Find those on a map and you get a pretty good idea why the area works so well. The zone I point out is along the Andrassy ut corridor which is a beautiful walk, but the M1 metro runs right under the street if you just want to make time. The M1 will connect you with all the above ground transportation you will want to use. Both the M1 and the Trams are fun and not merely transportation.

The area bounded by Deak Ferenc ter on the east, the river on the west, and the Erzsébet híd (bridge) on the south and the Chain Bridge on the north sort of defines the Tourist Hell of Budapest.

Okay, Tourist Hell is strong. This is the zone that exists for the sole purpose of serving tourists. Tourist restaurants, all the big name brand stores that you find in every tourist center, hundreds of souvenir shops and everything substantially more expensive than Budapest as a whole. It, in and of its self is a touris attraction, but nothing in the zone is of particular “Hungarian” or “Budapest” interest other than the river front which is quite beautiful.

The apartment you should be able to find within the zone I described for about 60 euro a night. Also significant to the area I suggest is that in the evening you can walk among the Hungarians, eat in Hungarian restaurants and drink in Hungarian wine bars; all at Hungarian prices. But I am terribly biased……..
Hungarian lesson:

utca = street
ut = boulevard or major road
ter = square

And this is just my personal preference and where we stay when in town 3 times a year. There are a lot of good places, just dependent on your preferences and taste.

For me, mid September to mid October is the absolute best time in Budapest.

For a cultural experience visit the Operett, Opus Jazz, Ruin Pubs, a working Synagogue, Bath House, Wine Bar, drink Palinka, hear a Klezmer concert, and explore District VIII.

Thank you this is good info! We definitely want to get an authentic experience while there and not focus our time on really touristy stuff that we could do anywhere (which is what I meant by 'cultural experience'...meant we want to experience Hungarian culture specifically). We will of course want to do at least some tourist things to see history and sights and such as they are, but not interested in going to malls or doing things we could do anywhere so it's nice to know the tourist zone you mention so we can avoid it. :)

The tourist zone, the Vth District is worth seeing, just not worth staying in. That's where the river embankment is and it's stunning at night.

This is going to sound a little extreme, but the best way to get a cultural experience is to function inside the culture. I get a lot of my medical work done here and hire a personal trainer to get some exercise in the mornings before sight seeing. Figure 30% to 60% of the cost in the US for high end clinics.

I have just returned for a week in Budapest (my first and certainly not my last visit). I have stopped using Airbnb where possible, as their admin fees are getting ridiculous for what they offer, plus their exchange rates are poor/abysmal. I had selected a place in Budapest on Airbnb and when I clicked to book, the price shot up by 40%! I find in Europe that Trip Advisor rentals are much better value for money and are just as safe and secure.

I rented a one bed place from the Trip Advisor website near Oktogon, in Jokai Ter - the location was fantastic, with lots of bars, restaurants and transport nearby, but the flat was very quiet at night.

Most of Budapest is within close walking distance or near to good public transport, so there is no need to move locations. A multi day travel ticket is about the cheapest that I have ever come across anywhere in Europe and is worth getting.

AirBnb, HomeAway, or ??? They are just booking agents. Pick a place you like then go hunting for booking agent that gives you what you want. For instance, there is a place at Jokai ter (square) that has in its description “Apartment is situated in a quiet court yard away from the hustle and bustle of the City night life”. Since that is a pretty unique statement you can copy it and put it in Google to see what else pops up. This particular apartment is listed with five different agencies (including AirBnb). I would choose the local Budapest company in this case (I have absolutely nothing to do with the apartment for the listing company). My nephew stayed in an apartment on Jokai ter a few weeks ago (not this one) and we visited. Not a bad little area. The only minor criticism is that there are so many short term rentals on the square that you are going to see fewer “Hungarians”.