As the authors explain in the 2010-04-22 version of their text, an explicit presentation was first obtained by Harer, using the method developed by Hatcher and Thurston. The latter defined a 2-dimensional simply connected polyhedral complex on which Mod(S) acts :

1. cocompactly (i.e. finitely many orbits of 0,1, and 2 faces)
2. with explicitely finitely presented vertex stabilizers (close to braid groups)
3. with explicitely finitely generated edge stabilizers

A general recipe then says how to obtain a finite presentation of the whole group Mod(S),
which Harer was the first to take to go through in all cases, obtaining a "somewhat unwieldy presentation".

Then Wajnryb managed to simplify Harer's presentation. Personally, I prefer Makoto Matsumoto's
beautiful presentation (obtained by simplifying Wajnryb's one, with the aid of a computer), where relations added to the braid ones are naturally indexed by some Dynkin diagrams (ADE again!)