Seems to have been Alembic's modifications of Jack Cassady's and Phil Lesh's Guild Starfire basses. Then they built Cassady's Alembic bass.
I love Alembic basses and have 2. But I think Les Paul had bass with active electronics; in the Les Paul Recording Bass, made prior to Alembic. However Alembic was the first to get them out there, and did it right. The Les Paul was made in 1969. Alembic #1 was 1972. I hope I am right here.
May be wrong, but I think Ovation tried to claim to be the first mass manufacturer of an active bass model.
I believe the Hofner 500/8-BZ had a built in fuzz the required an on-board battery (if someone knows different, please correct me). How do you want to define "active"? I've always thought the term referred to an on-board, tone-altering circuit which requires a power source. I'm no authority (all my basses are passive) so straighten me out if that's wrong. I keep all that stuff on the pedal board run by a Strymon power block because I hate missing with batteries.
If the battery was required to get any sound out of it I'd call it active. If the battery was only required to use the fuzz circuit, i wouldn't call it an active bass. That's misleading.
The Les Paul Bass and Triumph bass had low impedance electronics but I'm not sure they were powered i.e. active... I'm sure @Basvarken would know.
They're passive. No batteries required! The Les Paul Bass came with a special (external) transformer plug to go from lo-z to hi-z. And the Triumph has a built in impedance transformer. You can choose between lo-z to hi-z with the slider switch on the control plate.
The Les Paul Recording Bass had weird electronics, but it was still passive. If you count on-board effects as "active", the Gibson EB0-F in 1962 would be the first production model I can think of. Vox started adding onboard FX in 1967 to both guitars and basses, including a Treble/Bass Booster, which could be considered a form of active EQ. They also built a guitar-only model with a strings-triggered organ inside!!! Hofner started offering preamps in their guitars as early as the 1950s and they were later available for basses as well.
Similar experimentation was going on with guitars starting in the late 1960s: MCI – 1967-1988 MusiConics International, Waco Texas, famous for the guitorgan – modified Gibson, Ventura or Univox archtops. MCI, Inc., of Waco, Texas, was a distributor for the Daion Guitar Company in Japan. MCI, Inc., was active from 1977 until 1983, and in early 1984 the decision was made to totally restructure the company interest in music and undertake some major expansion programs in manufacturing, merchandising, and marketing. The first expansion was in the manufacturing and marketing of the company’s MCI steel guitars. Then, there was also the Vox V251 Guitar Organ
The Hofner is a bass preamp: Hofner Active Guitar Electronics - Fact File (link copied from @brianrost )