The first crossing (from Victoria to Tasmania) was done in April 1971 in slalom kayaks by 3 Victorian paddlers (Rod Harris, Ian and Peter Richards) from a club in Ivanhoe. See Wild magazune 118 - July - August 2010.
Then it was done by Cliff Wright and Colin Rose as part of their Sydney - Hobart paddle in early 1972.
A return crossing of Bass Strait from Tasmania to Victoria and back was done solo in 1982 by myself,
Four members of the Maatsuyker Canoe Club (including one young lady who had only been sea canoeing for three months) made a return trip in 1986/87.
Larry Gray, Graeme Joy, Rob Casamento and one other paddled from Tasmania to Victoria in December 1986.
Peter Treseder claims he did a 38 hour non-stop crossing to Tasmania, and then paddled back to Victoria in 1992? There appears
          to be serious doubts about this claim. See article by Peter Trebey. (You may be interested in this, about other claims by Peter Treseder)
Ken Miller, Graeme Albrey, and 4 other West Australians paddled from Victoria to Tasmania in December 1994.
John Wilde and 2 others did a similar trip in April 1995.
John Jacoby and 2 others went from Tasmania to Victoria in March 1997.
Matt Dieglutis and Wade Burnham crossed from Victoria to Tasmania in January 1998.
Karl Noonan crossed in February 1998 as part of his Sydney - Hobart paddle.
Wayne Langmaid, Andrew Linton, and a group crossed from Victoria to Tasmania in February 1998.
Matt O'Brien and Pete Lenehan went from Tasmania to Victoria in April 1998.
Stuart Trueman did a solo crossing from Victoria to Tasmania in 1998/99.
Steve Dineen and Lester Nation used a double in March 1999 to cross from Victoria to Tasmania.
The first all South Australia crossing of Bass Strait via Flinders Island was completed in April 1999 by Malcolm Hamilton, Phil Doddridge, Gordon Begg, Scott Polley, Tim Vogt and David Williamson
Four members of the VSKC did it in February 2000. Separate reports by Julian Smith and Ian Dunn
Some members of the Maatsuyker club paddled from Victoria to Tasmania in February 2001.
David Williamson crossed from Tas to Apollo Bay via King Island in Jan 2001.
Three NSW women (one had to pull out at Deal Island after a fall) paddled from Victoria to Tasmania in February 2001.
Vince Browning did a solo trip from Wilsons Prom to Little Musselroe in Jan 2003.
Hayley Shields (a Victorian) paddled from Port Welshpool to Bridport with three companions, in March 2003,
           then continued on down the east coast of Tasmania to Hobart, where she is enrolled at the University.
Ian Soutar and Dick Condon (to celebrate their 60th birthdays) paddled from Wilsons Prom to Little Musselroe Bay in February 2004.
           I understand that 2/3 of the paddle from Deal Island to Flinders Island was done in 40 knot winds - hairy stuff.
John Jacoby and Jarad Kohlar paddled from Victoria to Tasmania via King Island in November 2004.
Andrew Peacock and Cameron Webb paddled from Victoria to Tasmania in the summer of 2008.
Stuart Trueman did a nonstop crossing in Nov '08
Darren Smith and Nick Bladen crossed in a double in March 2009. (it is a large pdf file)
Mike Snoad did a solo crossing in April 2009, turning 65 on the trip - probably the oldest person to paddle across Bass Strait.
Mark Schroeder & Matt Bezzina paddled across in March, 2010.
Shaan Gresser is probably the first woman to paddle solo across Bass Strait in February, 2011.
Geoff Murray and five others crossed in March 2011.

Many adventurous paddlers are crossing Bass Strait these days, and I will endeavour to make these trip reports available online if they are sent to me.