In Surfer, it is possible to assign NoData values to grid nodes outside of multiple polygons, however these polygons will need to be combined into a single complex polygon, or else the entire grid may be assigned NoData values. This is a result of the area outside one polygon being included inside the other polygon, and vice versa. 

The area outside Polygon A includes Polygon B, and vice versa,
which causes the entire grid to be assigned NoData when using multiple polygons.

You can use the Features  | New Features | Combine command to combine multiple polygons into a single complex polygon, and then re-export the BLN to use in assigning NoData values to the grid. The exact steps are as follows:

1. Click Home | New Map | Base | Base, select the original BLN file and click Open. A base map is created showing the existing polygons.
2. In the Contents window, select the Base(vector) object.
3. In the Contents window, click the  button to expand the base layer, then select all the Polygon objects below the Base layer. Hold the CTRL or SHIFT key on the keyboard to select multiple polygons.
4. Click Features  | New Features | Combine. Now the polygons are combined into a single complex polygon.
5. In the Contents window, turn off the axes by unchecking them.
6. Click File | Export.
7. In the Export dialog, enter a File name, change the Save as type to BLN Golden Software Blanking (*.bln), and click Save.
8. In the Export Options dialog, on the Scaling page, make sure the Scaling source is set to Map.
9. On the BLN Options tab, make sure the Break apart compound areas option is not checked and Outside is toggled in the NoData areas section.
10. Click OK. 

The BLN file that is created will contain one complex polygon and can then be used to blank the grid file outside the polygon.

Updated April, 2020