In Subtitle Edit, right-click the subtitle event you want to reposition, then select “Alignment”.
HandBrake is a video transcoder. It's not designed for subtitle editing. 
Sarper Aman wrote:
You will only see position changes in SE if you use "mpv" as video player - see Options -> Settings -> Video player .
Note: Not all video players/converters will use this positioning.
I believe this is also possible with VLC as the playback engine. Although I could be wrong. I swapped it for mpv a long, long time ago.
Stepan Konev wrote:
The alignment tool mentioned above only applies to the SE videoplayer (for preview). If you save your output file in the SRT format, it would not contain any information about the subtitles position. SRT is just a plain text. You will have to adjust the subs position in the videoplayer that you use for playing the video.
Thus, you have two options: (1) embed the subs with adjusted position or (2) change the subs format and adjust the subs position.
I've completed quite a few projects in SRT format using SE. And I can assure you, that the soft subtitles followed my custom positioning without hardcoding them onto the videos. 
[Edited at 2024-03-26 11:53 GMT]