Let's step back and look at the reason for making a DV01 calculation first before answering the question;
The reason for making a DV01 calculation is to quantify what market movements has impact on the valuation of the trade.
Since the 'flat' forecast curve won't be affected by market movements the answer is (using pre-2008 methodology): The floating forecast curve.
After 2008 the discount curves became more important regarding the valuation as the previous standard to discount on the floating forecast curve (aka. IBOR-curve) was replaced by discounting on OIS (Overnight Index Swap)-curves or discount curves based on the collateral posted by the trade counter party.
In such a case DV01 would be calculated for the forecast curve, and for the discounting curve (which should be the same for both legs of the swap as long as both legs are in the same currency), resulting in two DV01 measurements.