My own field, ergodic theory, is relatively young in that some concepts now regarded as fundamental -- Kolmogorov-Sinai entropy, for example -- were not fully formulated until around 1960. Nonetheless there are a couple of old books still in use and receiving citations:
E. Hopf, Ergodentheorie, 1937;
R. Halmos, Ergodic theory, 1957.
If the 1960s are sufficiently long ago to constitute "old" then there are many old references in probability which remain in heavy use, for example:
P. Billingsley, Convergence of probability measures, 1968;
L. Breiman, Probability, 1968;
and one of the classics of the field:
W. Feller, Introduction to probability theory and its applications, 1950.
Outside my own field, a much-cited number theory text which no-one has yet mentioned:
A. Khinchin, Continued fractions, 1936.