My answer is not limit to K means, but check if we have curse of dimensionality for any distance based methods. K-means is based on a distance measure (for example, Euclidean distance)

Before run the algorithm, we can check the distance metric distribution, i.e., all distance metrics for all pairs in of data. If you have $N$ data points, you should have $0.5\cdot N\cdot(N-1)$ distance metrics. If the data is too large, we can check a sample of that.

If we have the curse of dimensionality problem, what you will see, is that these values are very close to each other. This seems very counter-intuitive, because it means every one is close or far away from every one and distance measure is basically useless.

----------

Here is some simulation to show you such counter-intuitive results. If all of the features are uniformly distributed, and if there are have too many dimensions, every distance metrics should be close to $\frac 1 6$, which comes from $\int_{x_i=0}^1\int_{x_j=0}^1 (x_i-x_j)^2 dx_i dx_j$

Here is the simulation for dimension from 1 to 500, the features are uniform distribution from 0 to 1.

    plot(0, type="n",xlim=c(0,0.5),ylim=c(0,50))
    abline(v=1/6,lty=2,col=2)
    grid()

    n_data=1e3
    for (p in c(1:5,10,15,20,25,50,100,250,500)){
        x=matrix(runif(n_data*p),ncol=p)
        all_dist=as.vector(dist(x))^2/p
        lines(density(all_dist))
    }
[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/b30KW.png