What is more important in Antarctica calving or basal melting of the ice shelves?
There are two studies published in 2013 that provide the most recent updates on this balance

Depoorter, M.A. and 6 others, 2013. Calving fluxes and basal melt rates of Antarctic ice shelves. Nature, 502, 89–92, (03 October 2013),  doi:10.1038/nature12567
Rignot, E. and 3 others, 2013. Ice-Shelf Melting Around Antarctica. Science, 341, 6143, 266-270, (19 July 2013), doi: 10.1126/science.1235798

The first study arrives at a calving loss of 1321 ± 144 Gt/yr year and a total basal mass balance of −1454 ± 174 Gt/yr (− = loss). The mass balance is the net of mass added by freeze on and mass lost by melting so the number does not reflect melting alone. The pure loss by melt is hence larger but from a balance perspective the net is what counts.
The second study indicates a calving flux of 1089 ± 139 Gt/yr and a basal melt of 1325 ± 235 Gt/yr. Although they report &quot;melting&quot; it is likely that they mean net balance as in the other study.
The two studies have used slightly different approaches to arrive at their results, both indirect, but over all the numbers indicate that basal losses are larger than calving and that the order of magnitude is the same.
In more detail, the studies show that calving losses are dominant from the larger ice shelves, the Ross and Filchner-Ronne. Melting dominate regions along the western Antarctic Peninsula and down along the coast of the Bellingshausen and Amundsen seas. For most of the East and West Indian Ocean coastlines losses are relatively small and equal between the two processes.
The bottom line is that these studies have changed the perspective from ice loss being predominantly through calving to now slightly dominated by basal melting.