With the exception of basic lands, are all Alpha, Beta, and Arabian Nights cards banned in sanctioned tournaments? Does that also include Deserts?
There is a special limitation on Alpha cards to account for their differently-cut edges. From the official Magic The Gathering Floor Rules:
102. Authorized Cards
Alpha cards (cards from the first print run of the basic set) may be used in decks containing non-Alpha cards only if all cards are placed in completely opaque sleeves and only if the sleeves could not be considered marked. If sleeves are not used, Alpha cards may be used only in decks that consist exclusively of Alpha cards.
There are no such restrictions on Beta or Arabian Nights cards however, which are (hopefully) less easily distinguishable by their shape from modern cards.
As rahzark's answer rightly suggests, individual cards may be restricted or banned in different tournament formats, but there's no prohibition against cards from older sets as a whole.
First of all, sanctioned tournaments have different formats, which allow for different cards. You can read more about all sanctioned formats here.
Most cards from those expansions (like Desert) aren't banned so you can play with them in Legacy or Vintage!