Chemistry - How is seawater alkaline?

The short answer to the title question is the ubiquitous presence of geologic and biogenic calcium carbonate $(\mathrm{p}K_\mathrm{a} = 9)$. The oceans as a whole can largely be thought of as residing on beds of calcium carbonate. The following are some excerpts from this Wikipedia page:

Eggshells, snail shells and most seashells are predominantly calcium carbonate.

Carbonate is found frequently in geologic settings and constitute an enormous carbon reservoir. Calcium carbonate occurs as aragonite, calcite and dolomite. The carbonate minerals form the rock types: limestone, chalk, marble, travertine, tufa, and others.

Calcium carbonate contributors, including plankton (such as coccoliths and planktic foraminifera), coralline algae, sponges, brachiopods, echinoderms, bryozoa and mollusks, are typically found in shallow water environments where sunlight and filterable food are more abundant. Cold-water carbonates do exist at higher latitudes but have a very slow growth rate.

Regarding your question

Doesn't a compound need an hydroxide ion to be a base?

Here you go:

$$\ce{CO3^2- (aq) + H2O (l) <=> HCO3- (aq) + OH-(aq)}$$

Although calcium carbonate is only sparingly soluble $(K_\mathrm{sp} = 3.3×10^{−9})$, as stated above the oceans are simply rife with it.