When using box-select in the Video Sequencer, only whole strips can be selected. It is currently not possible to use it for handle selection.
There is an add-on to address this, but ideally this should be supported in Blender itself: https://github.com/Salatfreak/VSEBoxSelectHandles
The following animation from above's link demonstrates the desired behavior:
(It's not recommended to use the Add-on code as a reference, as the C implementation would have to be quite different.)
Once this is implemented, we can check on some design questions, like what the default should be and which shortcuts to use. Since these things can easily be changed in the keymap, it's not holding back the implementation work.
This should be implemented as an operator option, which should be named Include Handles for consistency with a similar option in the Graph Editor box-select operator.
Besides that, the Graph Editor operator doesn't serve as a good reference either.