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 running not algorithm, we can check all distance metrics for all pairs in data. 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 the distribution of these distance metrics are closee. If all of your features are uniform distribution, and if you have too many dimensions, every distance should be close to $\frac 1 6$.
Here is the simulation for dimension from 1 to 500, every feature is uniform.
