Visit the LJ homepage, click on the link to your friend page and check that it worked.
That's it. No muss, no fuss, no need to open a support request.
Special note for users with S1 Journal layouts If the above instructions didn't work, go visit the Journal Style page and check to see if you're using an original S1 journal layout. Apparently there's a bug right now preventing user with S1 journal layouts from using the original /friends page. LiveJournal deprecated S1 over a decade ago and while they've been pretty good about keeping it working I've got a feeling this might be the incident that prompts them to pull the plug. So you can either wait for them to maybe fix the bug, or it might be time to explore some of the newer S2 layouts. If you do switch be aware that there is no longer an option to switch back to your S1 layout.
Other display options that affect seeing the new design There are a few other options that affect whether or not and where you'll see the new LiveJournal design. And as a reminder the overall new design, the friends /feed, and the new Air journal layout all match each other visually, but are affected by different options.
We've already talked about the /feed page up above. The default colors for that match the white, blue, and grey of the new site design but can be altered by clicking on the gear icon on the left side of the feed. For instance, [I have a nice autumn harvest color scheme on mine.]
The new site design itself. In the old days we called it a site scheme (Dystopia, xColibur, Vertigo, or Horizon) but now it's just "the new design." This is the white, blue, and grey you see on all the site pages on LJ (your profile, post an entry page, memories, the homepage, etc...) There's currently no way to change these colors so you're stuck with these default colors. a) If you're using the new design and want to switch back to one of the old site schemes visit Display Settings, check the box at the bottom of the page labeled "Set old design version" and click Save. b) If you're currently using one of the old site schemes and want to switch to the new design just click on the "Switch to new version" link at the top of any site page.
The new Air journal layout is now an option for users to select for their journal. All new users now start out with this layout, but older users can choose to switch to it as well. It matches the new design and the /feed page in look and feel and again, the default colors are white, blue, and grey with no way to change them.
Mobile devices. There's a new setting that affects whether users on mobile devices (phones, tablets, etc...) will see your regular journal layout or the new Air layout when viewing your journal directly. By default all personal journals and communities were switched to showing Air to mobile users but there is an option to go back. a) To display your chosen journal layout while on mobile devices visit Display Settings and check the box in the middle of the page labeled "Use my journal's style when viewed on a mobile device" and click Save. b) Conversely, to display the new Air while on a mobile device layout uncheck that box and click Save.
If you opt-out of using Air for mobile devices there's still another distinction: Basic & Plus journals will only show their chosen styles to themselves while logged in. Paid & Permanent journals will show their chosen styles to all users, logged-in or otherwise.
* I found it via an entry I saw on the homepage, but unfortunately it was on my iPad I don't have a way to find it again for credit.