The situation you describe will arise as a result of one of these two scenarios:
- The column you're referring to is the column of 1's which is added to your matrix of covariates so that your linear regression has an intercept term.
- The column is a different column than the previously-mentioned column of ones, giving you two columns of constants [****].
For Scenario 1: skip that column, standardize all the other columns, and then run the regression as you normally would.
For Scenario 2, however, you'll have to get rid of that additional constant column entirely. In fact, regardless of the question of Standardization, you'll never be able to run the regression with two constant columns since then you would have perfect collinearity. The result is that even if you try running the regression, the computer program will spit out an error message and quit halfway through [Note: this is because an OLS regression requires the matrix X'X to be non-singular for things to work out correctly].
Anyway, good luck with your, um, regressing!
[****] Just to clarify: What I mean by "two columns of constants" is that you have one column in which every element is '1' and a second column in which every element is some constant 'k'...