Google ceased support for Picasa software. They insist on using the Google Photos for everything related to photo organizing or manipulation, but they also omitted the offline part.
However, if you have the software installed, use a USB flash drive and copy the installation folder of Picasa to the USB drive, then plug it on the new computer and paste it in the same location where it was previously stored on the older one. When it's launched from the new location, Picasa will create a new database file. You can then add the images as you please.