Robinson clocked the 15th fastest time overall in the first run and improved one place with the 10th fastest time in the second run.
But her combined time was 3.93 seconds slower than Bassino who notched up her third win in four races this season ahead of Frenchwoman Tessa Worley and Switzerland's Michelle Gisin.
Kranjska Gora was a late change to the schedule with a lack of snow forcing this weekend's races to be moved from Maribor to the resort higher in the mountains nearer the Slovenian border with Austria.
A year ago, Robinson powered to her second victory of the Covid-shortened season with her second run nearly half a second quicker than her nearest rival and a full second over everyone else.
Robinson has struggled to come close to last season's heroics so far this season but will be relieved to at least finish both runs after failing to finish the last two completed Giant Slaloms in Courchevel in December.
Bassino has set the benchmark in the discipline this season winning both Solden, the first race in Courchevel and now the first race in Kranjska Gora.
The second race is tonight.