Testing for drift in random walk How would you test whether a unit root I(1) process has a statistically significant constant?
To my understanding, the AR(1) regression in the levels is statistically spurious, and hence we cannot make inference on the constant from there.
Thank you.
 A: Your title and the first line do not seem to match exactly: a drift in a random walk implies a linear time trend added to a random walk rather than a constant level added to a random walk, but perhaps it's only me misreading. I will consider the former case.
If you are certain that the process has a unit root, you can legimately difference it. Then the drift in levels becomes a constant in the first differences. You may regress the differenced series on a constant and use the regular $t$-test. The standard regression output will immediately show whether the constant is significantly different from zero. 
If you are uncertain whether the process has a unit root, there is an option to include a drift term in the augmented Dickey-Fuller test. Again, the statistical significance of the drift term will normally be included in the standard output of the test regression.

A side note: if you did not care about testing the statistical significance but only were interested in the coefficient value of the drift term, perhaps you could get away with regressing the data in levels on a linear time trend. The idea is that a linear time trend dominates a stochastic trend (a random walk); if there really is a time trend in the data, it will be "visible" even in presence of an added stochastic trend. The argumentation would be similar to estimating a bivariate regression for a pair of cointegrated variables, as is done in the first step of the Engle-Granger cointegration testing procedure. The common stochastic trend will be "visible" even in presence of added stationary components. However, this side note is provided only for curiosity and does not address your problem directly.
