The opponents of phylogenetic systematics ("cladism"), that is the proponents of non-monophyletic taxa, usually seem to have one of two views of the current situation in systematic biology:
Some of them feel that they are part of a silent majority that is increasingly asserting itself, that there is a sea change, a growing realisation that cladism doesn't make sense and that non-monophyletic taxa need to be accepted. They perceive an increasing number of scientific publications arguing for their position.
Others feel that they are a suppressed minority, whose warnings about the wrong turn that systematics has supposedly taken over the past few decades are ignored and unfairly kept out of the major journals in the area.
These two perspectives are not entirely mutually exclusive - it could be possible that a cabal of influential editors is maintaining the dominance of cladism but that their days are numbered as the next generation moves in - but they still make for a very different perspective on where things are going at the moment. Does phylogenetic systematics have a clear hegemony, and will it last?
My own feeling is that it depends on what you count, what you particularly care for.
To me it feels as if cladism has achieved clear hegemony as far as the principles of classification are concerned: the major journals and the leading experts in the field are near-unanimous that the goal is to make all supraspecific taxa monophyletic. Yes, there are some vocal opponents, no doubt about that. But they are a minority, and it does not seem to be growing; indeed it seems to include very few young systematists.
There are several pieces of (admittedly partly anecdotal) evidence that lead me to this view. One is just what you get out of looking into all the publications and conference talks that are coming out; pretty much always the authors and speakers either express satisfaction that a group is monophyletic or they suggest its re-circumscription to make it so. It is just standard practice. For example, I cannot remember ever having peer-reviewed a research paper whose authors suggest to make a group non-monophyletic. And I run into enough people, especially among zoologists, who are downright incredulous that we are even still discussing that issue.
Whether it will stay that way depends, I believe, partly on how well students are taught the details. There are, unfortunately, a lot of colleagues out there whose understanding starts and ends at "stuff should be monophyletic". On the one hand that leads to over-reach, such as talk of making sexually reproducing species monophyletic, as if there was any phylogenetic structure between groups of happily interbreeding individuals. On the other hand, there are then those who do understand that there is no phylogenetic structure inside a species-lineage but jump from there to the conclusion that therefore cladism is nonsense - because nobody ever told them that even the founder of cladism himself would have considered the idea of making a single species monophyletic to be incoherent. Still, this is nothing that a few words couldn't clear up.
However, depending on your definition of cladism it may also include a preference for parsimony analysis. As I have mentioned before, there are quite a few old school cladists who reject model-based and Bayesian analyses for theoretical reasons. Myself, I am a methodological pragmatist, but if one considers an exclusive reliance on parsimony analysis to be a core part of cladism then cladism has definitely lost that battle.
In the past ten years or so Bayesian analysis has gained ascendency in systematic biology. There are many people who look down on parsimony and consider it old-fashioned and unsophisticated. I will be the first to point out the many shortcomings of Bayesian approaches - steep learning curve for beginners, numerous often unjustifiable assumptions and priors, black box feeling when using these tools, often ridiculous computing times - but for many research questions the available parsimony methods just don't cut it.
Just for starters, if you want to do an analysis of shifts in diversification rates, or use certain algorithms of ancestral area reconstruction, or just have a chronogram to figure out when something may have happened, you need an ultrametric, time calibrated tree. You will not get one from parsimony, and that is that. So it is very understandable that a whole generation of students and postdocs interested in these kinds of research has left parsimony behind.
Finally, there is the question of rank-free classification. Of course, not every cladist is a PhyloCoder, but it does seem to me that one of the insights that follow logically from phylogenetic systematics is that ranks above the species level are ultimately arbitrary and meaningless. Diversity is organised in nested clades, but you cannot meaningfully compare a genus of beetles and a genus of algae; there is no genus-ness in nature. What is more, classifying species that have existed in past eras leads to rather bad tensions when using the pre-Theory of Evolution Linnean system. So the fate of the primary vehicle for a rank-free classification is also of interest in this context.
Quite apart from the problem that the PhyloCode does not really appear to have found a widely acceptable solution for species names yet, the problem as I see it is that the best time window for a push towards a rank-free classification has passed. I work a lot with biodiversity databases - GBIF, Genbank, and so on - and all these databases have been constructed to use the ranked Linnean system. Yes, classes, orders and suchlike do not really have any meaning and they are terribly clunky. But I am afraid they are here to stay for the foreseeable future, because now we have those databases nobody will be willing to do everything over. The best chance would perhaps have been to convince people when those databases were first developed that a rank-free system would make the management of biodiversity information, the accommodation of taxonomic changes, and the addition of new groups easier, but that apparently didn't happen.