Given that Batanes is a far-flung group of islands, the internet connection here is quite limited and slow. Some hotels have it but they will often ask for an extra charge to use it; if you don't want that, you can buy a Smart or Globe SIM card but they will only have LTE signal strength. There are also some computer cafes found in Basco with a decent internet connection, primarily used by the younger ones for gaming — but still, those are worth a shot.