Are you able to go into "Settings" on the phone and then choose "iCloud" and finally make sure that the option to save "Safari" files there is on?
If you can do that, then you should then be able to access your "iCloud" from the computer and choose where you want to "Sync" those bookmarks to on the computer.
If the first step doesn't work, then there is something wrong or missing on your phone "Settings" or if you can't get the info to "Sync" to the "iCloud" account (and show up there), you would need to check with someone who deals with apple devices/programs. I am Android and PC only.
Now if you can do those steps, but can't log into your "iCloud" from the computer, then you would have to contact them about why you cannot. The same goes for if you can access it, but cannot import the file to the computer.
Those are the only options on how to do it that I know of. Anything different (for the Apple devices) you will have to talk to an Apple person about.