The judiciary branch enforces it to some extent by 18 U.S. Code § 2383 (thanks to @Nate Eldridge for pointing this out):
Whoever incites, sets on foot, assists, or engages in any rebellion or insurrection against the authority of the United States or the laws thereof, or gives aid or comfort thereto, shall be fined under this title or imprisoned not more than ten years, or both; and shall be incapable of holding any office under the United States.
The more indeterminate question is whether Congress itself can make the determination of insurrection and bar someone from running. I'm unsure what is the answer to that, but former officials were barred from holding any other office after being impeached and convicted, which is possible due to a different provision (Art. I sec. 3 cl. 7) that explicitly allows disqualification (by the Senate) in those circumstances.
I think it's untested whether Congress by itself can merely bar someone in particular from holding office, as this may butt against the constitutional prohibition on bills of attainder. The case law on that is pretty convoluted, and some tests depend on whether some measure is considered a punishment or not, and whether punishment is the "main purpose" of the law. User phoog has now written an answer on law SE that (fairly coherently, imho) argues that the answer to that is "no, Congress can't do that by itself"; as user grovkin also pointed out over there, Alan Dershowitz (who defended Trump before) has been arguing along the same lines.
On the other hand, Reuters also covered this issue, and mentions a precedent in re direct application of Congress of the 14th, namely Victor Berger:
Under congressional precedent, only a simple majority of both chambers is needed to invoke this penalty. Congress can later remove the disqualification, but only if two-thirds of both houses vote in favor of doing so.
In 1919, Congress used the 14th Amendment to block an elected official, Victor Berger, from assuming his seat in the House because he had actively opposed U.S. intervention in World War I.
According to Wikipedia, Berger's actual conviction for espionage was overturned but his bar by Congress was not, although the reversal by SCOTUS came a bit late for Congress to consider it during Berger's actual term in office. Basically he sat out his term 1918-1920, but after SCOTUS overturned his conviction (on January 31, 1921) and he was reelected again in 1922 (and '24 and '26), I see no mention of Congress not seating him again, so I assume they relented. The fact that Berger's seating was not opposed from 1923 onwards (or even subject to a vote in the House thereafter) is explicitly confirmed
in another source, but I'll omit quoting from that since it's written in fairly hagiographical tone.