The city of Luxor on the East Bank has several Mini-bus / Bus routes used mainly by locals.
Taxis are plentiful, and reasonably priced, and since the government has decreed that taxis older than 20 years will not be re-licensed, there are many modern air-conditioned cabs. Recently, new roads have been built in the city to cope with the growth in traffic.
Upon request:
Traditionally, however, river crossings have been the domain of several ferry services. The so-called 'local ferry' (also known as the 'National Ferry') continues to operate from a landing opposite the Temple of Luxor. The single fare (June 2008) is 1 L.E. - one Egyptian Pound - per passenger for foreigners.
This ferry is mainly used by the locals although a number of foreigners do use it. The sites on the West Bank are further than you think and you will need transport--taxi drivers often approach ferry passengers, and it is recommended that a fare be negotiated ahead of time. There are also local cars that reach some of the monuments for 25 piaster, although tourists rarely use them.
Alternatively, motorboats line the East Bank of the Nile all day providing a quicker, but more expensive (5 L.E.), crossing to the other side.