Other way to fix this is probably is to disable antialiasing in photoshop when saving
Update : you must save in png or tga or tff or any raw format , but without antialiasing , if you cant find the right setting you can google it , and here is not a photoshop forum and i cant explain more , and for resizing i suggest to use image > image size drop down ...
To fix the source of your problem, you should check your resizing workflow as @J.Doe suggested. Check your resizing mode/ interpolation and your source/target resolution in your external image editing tool.
To fix a resized texture, go into UV Editor and turn on snap to pixel. This way the UVs cannot cut Pixels in half and generate this sort of bleeding.
Hello and welcome to BlenderStackExchange,
For me using the latest UPBGE 0.3 release I oddly get the opposite problem;
Cube.001 is disappearing after runtime because Blender is automatically deleting as it wasn't originally created in Blender's viewport.
For me, unselecting the Undo at Exit setting prevented Blender from removing the newly created object ...