The intense red tide blooms (Karenia brevis) in the Gulf of Mexico are easy to detect with normal color photography because they are red (top image). The green algal blooms (lower image) are also easy to detect when they are intense.
Detecting these blooms is not so easy when they are less intense, when the water is muddy, when more than one species is present, or in shallow water where the bottom reflectance alters the color. The link provided by @MaggPi suggests that an NDVI-like index based on the difference between reflectance at 662 nm and at 680 nm is effective for detecting red tide. Expensive narrow-pass filters and a two-camera system would be required to capture these photos yourself. Satellites take images at these wavelengths every day.
A DIY single-camera NDVI system could probably highlight intense blooms in clear water. To detect red tide, which reflects lots of red light because the dinoflagellates use more blue light for photosynthesis, a blue filter might work better (these cameras use blue light to represent the color of light absorbed by plants for photosynthesis). To detect a green algal bloom, a red filter might work better because red light is used by those cameras to represent the color of light absorbed by the plants.
Too bad the editor for Answers is still not working.