Many studies, observations and measurements indicate that a great deal of the water recharges from the eastern highlands (Great Dividing Range). This answer focuses on Governmental reports (including the CSIRO)
According to the CSIRO report Hydrostratigraphy, hydrogeology
and system conceptualisation of the Great Artesian Basin (2012), the Great Artsian Basin consists of complex geological and hydrogeological features. This extensive study of the basin indicated no recharge from Papua New Guinea, stating that the recharge areas as being from folded Mesozoic rocks surrounding the basin in the central and southern parts, and Cenozoic aged recharge areas around the Gulf of Carpentaria in the north.
The recharge areas of the Great Artesian Basin are, according to the Australian Government Department of the Environment is mainly from rainfall on the highlands (near to the western margin of the Great Dividing Range) to the east of the Basin (with some from the western margin), as shown in the image below (from the Dept webpage):
Source:after Habermehl and Lau (1997). The shaded patterns broadly represent the
recharge area; arrows represent modelled flow lines after Welsh (2000). Dashed
lines represent spring clusters updated from Habermehl.
One major observation is that the top of Cape York is indicated as a recharge area, however, the CSIRO report details that this is recharge area is the Coen Inlier, just to the south of the shaded are. Also noted is that there is discharge into the Gulf of Carpentaria in the north - according to the CSIRO, the basin extends into the Gulf of Carpentaria. Although in all 3 reports there is no mention of recharge through the Torres Strait, the most any such discharge could influence, is the part of the basin in the Cape York/Carpentaria area.
A cross-section of a section of the basin is show in the image below (from the Queensland Government WetlandInfo webpage, clearly modelling the recharge areas (for the most part) being from the highlands on the eastern margin of the basin:
Habermehl MA and Lau JE 1997, Hydrogeology of the Great Artesian Basin (Map at scale
1:2500000), Australian Geological Survey Organisation, Canberra
Welsh WD 2000, GABFLOW: A steady state groundwater flow model of the Great Artesian
Basin, Bureau of Rural Sciences, Canberra.