I am considering two time series and I would like to to a VAR regression between them. The ADF test rejected stationarity in only one of them, so the time series would be I(0) and I(1). I understand the two processes can not be cointegrated since one of them is I(0).
If you are fairly sure that $Y_1$ is I(1) and $Y_2$ is I(0), take the first difference of $Y_1$ and model $(\Delta Y_1,Y_2)$ using a VAR. There is no point in using a VECM, as there is no cointegration and thus no error correction term.
There are a number of single-equation and panel unit root tests to check whether or not a variable is (non-)stationary. I suggest running at least KPSS in addition to ADF. Not all tests are created equal and some might be more powerful than others, especially in the case of near-unit-root processes.
As regards cointegrated VAR, it makes no sense if you have an I(1) variable and an I(0) variable, but given that you are not sure whether or not this is indeed the case, a Johansen VECM might throw light on this as well, because no cointegration in a Johansen test can be an indication of precisely what you are talking about, i.e. some variable(s) in the system are not integrated. Then you can proceed further.
