For anyone who is wondering about this question still, i will clarify - Volatility clustering does not at all imply that the series is non-stationary. It would suggest that there is a shifting conditional variance regime - which may still satisfy constancy of the unconditional distribution.
The GARCH(1,1) model of Bollerslev is not weakly stationary when $\alpha_{1}+\beta>1$, however it is actually still stricktly stationary for a much larger range, Nelson 1990. Further Rahbek & Jensen 2004 (Asymptotic inference in the non-stationary GARCH), showed that the ML estimator of $\alpha_{1}$ and $\beta$ is consistent and asymptotically normal for any parameter specification that ensure the model is non-stationary. Combining this with the results of Nelson 1990 (all weak or strict stationary GARCH(1,1) models have MLE estimator as consistent and asymptotically normal), suggests that any parameter combination whatsoever of $\alpha_{1}$ and $\beta>1$ will have consistent and Asymptotically normal estimators.
It is important to note however that if the GARCH(1,1) model is non stationary, the constant term in the conditional variance is not estimated consistently.
Regardless, this suggests that you do not have to worry about stationarity before estimating the GARCH model. You do however have to wonder whether it seems to have a symmetric distribution, and whether the series has high persistence, as this is not allowed in the classical GARCH(1,1) model.
When you have estimated the model it is of interest to test whether $\alpha_{1}+\beta=1$ if you are working with financial timeseries, since this would imply a trending conditional variance which is hard to immagine being a behavioral tendency amongst investors. Testing this however can be done with a normal LR test.
Stationarity is fairly misunderstood, and is only partially connected to whether the variance or mean seems to be ocationally changing - as this can still ocour while the process maintains a constant unconditional distribution.
The reason you may think that the seeming shifts in variance may cause a departure from stationarity, is because such a thing as permanent levelshift in the variance equation (or the mean equation) would by definition break stationarity. But if the changes are caused by the dynamic specification of the model, it may still be stationary even though the mean is impossible to identify and the volatility constantly changes. Another Beautiful example of this is the DAR(1,1) model introduced by Ling in 2002.