Not really. It has a deeper, richer quality than those cartoons. Filmation had Ted Knight doing the voice (he did most of the villains), and it wasn't quite as deep. I can't point to one actor and say it's inspired by that, but it is a combination of several influences, such as game show hosts/announcers, radio actors, cartoon voice actors, etc Really, it's a different kind a lunacy than Mark Hammil portrayed, especially since the tone was lighter.
If you've never seen the Filmation 60s cartoons, they are so much better than the Super Friends or the later New Adventures of Batman. They actually got to hit the villains, which led to uproars from watchdog groups, leading to much tougher rules about violence on Saturday morning cartoons. Looney Tunes and Tom & Jerry still got away with it for a while, in the 70s, but not Hanna-Barbera or Filmation. By the 80s, nobody could even throuw a punch, including Bugs and Daffy, except the Japanese imports, like Battle of the Planets (Gatchaman) and even they had the worst violence cut out.
"Fortunately, Ah keep mah feathers numbered for just such an emergency!"

Jerry Robinson, Bill Finger and Bob Kane based the Joker's appearance on actor Conrad Veidt in the 1928 movie The Man Who Laughs (about a man who's face has been disfigured into a permanent grin).
Image of Conrad Veidt in The Man Who Laughs: http://1.bp.blogspot.com/Inv0xFvLFGA/TAvphrktdgI/AAAAAAAADD4/D1jkdlz cUo/s1600/veidt+rit.png.
Really doesn't answer the OP's question seeing as
The Man Who Laughs
is a silent movie, and Conrad Veidt is a German actor who most definitely doesn't sound like The Joker in
Batman: The Brave and the Bold
. Also, the similarities between The Joker and Gwynplaine (The Man Who Laughs) ends at the physical resemblance as Conrad Veidt's character is the brooding protagonist who is stuck with a permanent smile when inside he's crying most of the time.
The Joker's voice in this show is probably most directly based on the old Batman cartoons, I would guess.
"There are times when I look at people and I see nothing worth liking."

I think the jokers voice in this series was based on the voice from the 60s cartoon.
When TVD brings Lucy and Sheila back my life will be complete.

Jerry Robinson, Bill Finger and Bob Kane based the Joker's appearance on actor Conrad Veidt in the 1928 movie
The Man Who Laughs
(about a man who's face has been disfigured into a permanent grin).
Image of Conrad Veidt in
The Man Who Laughs
:
http://1.bp.blogspot.com/Inv0xFvLFGA/TAvphrktdgI/AAAAAAAADD4/D1jkdlz cUo/s1600/veidt+rit.png
.
Burn the land and boil the sea
You can't take the sky from me

• Firefly theme