Incidents of tetanus were a problem during World War I, but barbed wire wasn't really the big reason for it. It was simply a matter of the environment, with soldiers developing tetanus due to deep-penetrating shrapnel wounds while fighting in muddy ground.

As you note, there was no vaccine at that point, but the incidents of tetanus did see notable development in its treatment during the conflict. Although it had been suggested decades before that anti-toxin serum could be injected soon after contact and provide effective treatment of tetanus, at the time Shibasaburo Kitasato and Emil von Behring's paper garnered little notice. Von Behring would go on to win a Nobel Prize, but for other work, on diphtheria, which was simply a much bigger deal compared to tetanus, which was a very rare and uncommon disease for the most part. World War One would be the first time that their suggested treatment really came to the forefront and saw any real use.

As already noted, World War I brought about conditions quite conducive to tetanus. Barbed wire wasn't the big deal, but rather deep shrapnel wounds. Especially in the early months of the war when the fields still had a nice spread of manure everywhere. The German 15th Army Corps, for example, reported 1,744 cases in the first three months of the war - roughly 6% of all wounded - and with a mortality rate of over 75%. British military hospitals reported even higher, with 8.6% of wounded developing tetanus infections, and likewise recorded very high mortality rates for those infected, although it varied greatly by hospital. This isn't to say that barbed wire never caused tetanus, but the focus of literature is on the more seriously wounded, as this was where the gravest threat lay.

Enter, stage right, the anti-toxin serum described as early as 1890. Recognizing the threat of the disease quick swiftly, it quickly became policy on both sides to administer the serum to wounded soldiers immediately as a preventative measure if it appeared the wound was infected with dirt or manure. The impact was almost immediate, with tetanus cases in hospital taking a steep decline. Millions of doses were administered during the war, and the serum saved many, many lives.

Even administering it was hardly a guarantee though. British testing pointed to a mortality rate that dropped from 87% to 38% which was a massive improvement but still showed how much of a threat tetanus remained. In turn some rather unethical French testing done with wounded German POWs - self-justifying the un-dosed control group due to supposed lack of enough doses - saw 18 undosed soldiers develop tetanus, and only one of those who received the serum do so. It ought to be noted though that the far from sterile conditions it was administered under likely was the cause of some small number of deaths from fatal serum anaphylaxis. Von Behring, who was a German doctor, continued to work at this problem and improve the serum, and would be awarded the Iron Cross, 2nd Class for his work.

The serum was not a vaccine though, conferring no long term immunity, with testing at the time showing it to be effective for roughly ten days time from the injection, requiring repeat injections weekly for seriously wounded men. As noted with von Behring, work to improve the effectiveness and reduce the likelihood of allergic reactions continued throughout the war, as tetanus continued to remain a serious threat for wounded men. A vaccine didn't come about during the war, but it would be hard to say that tetanus being suddenly thrust tot he forefront was anything less than a lighting rod in terms of attention to its development, not to mention simpy a better understanding of the disease. Writing in October, 1918 on where matters stood, Louis Bazy summarized matters in The Lancet thusly:

In short, it may be said that the war has brought us the following ideas :-1. Antitetanic preventive serotherapy is efficacious in the immense majority of cases. 2. When it facts incompletely it so modifies the course of tetanus that it has created new forms of the disease, unknown before its use was general. 3. The study of the checks to serotherapy ought to lead us (a) to use the serum in a more rational way ; and (b) to know how to complete its action by that of an antitetanic vaccination. These are some points that it is proposed to consider.

He discusses further the potential for a vaccine and how it might be brought about, and although it wouldn't be until several years after the war that a proper vaccine was developed, it is clear enough how well the groundwork for its development was laid during the conflict. But of course there is some irony then that once developed, its use was seen as minimal for some time still. With the war over, the primary need was of course gone, and due to the highly reactogenic nature of the early versions, it wasn't something to be commonly administered, more likely to harm the patient than they were to contract tetanus in most circumstances.

It wouldn't be until World War II that things changed. Again as noted, tetanus was seen as a very rare disease in civilian like, and it was the dirty circumstances of war that was the primary driving force, and the United States Surgeon General's Office had conducted interwar testing on how to improve the vaccine (removing "peptones"). The first large orders were for the US military which in turn was the first to administer it in mass quantities, and the improvements were so notable, and adverse effects so few, that the successful administration to the US military was enough justification for the American Pediatric Association to begin recommending mass vaccination beginning in 1944.

Sources

Bazy, L. (1918). "WHAT THE WAR HAS TAUGHT US ABOUT TETANUS". The Lancet, 192(4964), 523–526.

Hoyt, K. "Vaccine Innovation: Lessons from World War II". J Public Health Pol 27, 38–57 (2006).

Kantha, S. S. (1991). "A centennial review; the 1890 tetanus antitoxin paper of von Behring and Kitasato and the related developments". The Keio Journal of Medicine, 40(1), 35–39.

Shanks, G. D. (2014). "How World War 1 changed global attitudes to war and infectious diseases." The Lancet, 384(9955), 1699–1707.

Wever, P. C., & van Bergen, L. (2012). "Prevention of tetanus during the First World War". Medical Humanities, 38(2), 78–82.