Both are fully adequate, exf3 a bit more critical as a refuting attempt, and branches into several indipendent defensive setups. However black has much more than two challenging ideas, 4...c6 is another strong move (the strongest according to some sources, exf3 is delayed to a more favourable moment and white can't retake with 5.fxe4 due to the strong 5...e5 counter). Another fully adequate defence is the countergambit 3...e5. A french/caro player can just traspose to his favourite line with 2...e6/2...c6.
How to handle the Blackmar-Diemer Gambit

The example above is the Ryder Gambit. However sound you consider the Blackmar Diemer to be, Ryder is generally considered less sound by everybody. White might have enough initiative for one pawn, but certainly not two. That said, BDG players are usually also fluent with the Ryder, and might trot it out from time to time if they think they can make you stumble with it. I play BDG a lot, but I would never bother with the Ryder.
4. f3, Bf5 is the Vienna Defense. I actually think this is one of the most critical defenses Black can posit against this opening. White can either take the pawn 5.fxe4 Nxe4 6.Qf3 with a line that has been analyzed out to move 20 to follow. Alternatively he can play 5.g4!? the Hari Kiri attack, regaining his pawn in the center but at the cost of having an ugly looking pawn structure.

How do you handle the Blackmar-Diemer? 1) Using a stick, scrape the opening into a paper bag. 2)Staple the bag shut, then wrap the bag in aluminum foil. 3)Put that mess in a second, larger paper bag and drive it to a land fill. 4)Dump the bag and drive home at high speed. 5)Learn a decent opening
FYI:
As of Jan. 1, 2017 the EPA has banned the Blackmar-Diemer Gambit from all landfills. Toxins from the BDG are known to leach into groundwater and cause a health hazard. It's believed this is what killed Pillsbury, Trotsky and Nietzsche.
lol
I faced the Blackmar-Diemer Gambit in live chess yesterday, and afterword I found out black has two main tries to accept the gambit: