For the Requiem to have its full effect a pure, silvery soprano is needed in
Ihr habt nun Traurigkeit. And of course a great chorus and a sympathetic conductor. The baritone has more to do than the soprano (like in Fauré’s Requiem) but just like that work lives or dies by its
Pie Jesu movement, the emotional and devotional crux of the Brahms Requiem lies in that soprano solo. It’s the one piece of text that deals about death in an intimate, personal way. The other texts are more generic in that they treat death as a passage, a state, the gateway to the Great Beyond - conventional stuff suitable for a congregation of believers rather than the individual soul seeking spiritual comfort.
Of course it’s not enough to have a Gundula Janowitz or Elisabeth Schwarzkopf to sing in seraphic tones. In these particular recordings both Karajan and Klemperer don’t avoid some pitfalls: gooey religiosity (HvK) or severity bordering on austerity (OK). Karajan is better in his two subsequent recordings. Janowitz herself would do better under Haitink. There’s also the magnificent Margaret Price with Andre Previn or Elizabeth Watts under Nézet-Séguin. All three share the characteristic of adopting very slow tempi that make the experience rather depressing if one is not predisposed to equate slowness with profundity. The Kegel mentioned by Jo498 is a very good version - I even like the soprano

. Levine in Chicago is also very good - one of the best among easily available versions, in the box with the symphonies (superb).
Surprisingly the Requiem was never recorded (AFAIK) by the likes of Jochum, Böhm or Bernstein, who played the symphonies many times over throughout their careers. Walter is excellent but the sound is not modern.
Best for me though are two little-known, recentish versions offering fast speeds and luminous textures (orchestral and choral): that of Jan Willem de Vriend in The Hague :

and, best of all, this one at budget price, with a radiant Juliane Banse :
