Salicylic acid (sometimes referred to as BHA) does not increase skin's sensitivity towards the sun. You still should be using a sunscreen most day of the year, whether or not you are using a salicylic acid, but no extra precautions are needed for this exfoliating ingredient. This sets salicylic acid apart from another popular category of exfoliating acids - AHAs, and glycolic acid in particular that does increase skin's sensitivity to the UV light. This is why it is best to use glycolic acid in the evening and avoid sun exposure for a few days altogether after using glycolic acid peels in concentrations above 5%.
There is no harm in using salicylic acid in the morning even if you plan going outside during the day. Just make sure to use a sunscreen as usual, and do not overdo it with the exfoliation as thinner top layer of the skin offers less physical barrier for the sun rays.
Here is an interesting piece of skincare trivia: salicylic acid can actually have a small photo protective effect on the skin. Two clinical studies have showed that salicylic acid, when applied shortly before exposure to UV radiation, decreased the sunburn. It is important to emphasize though that in these experiments, a relatively high concentration of salicylic acid was applied in large quantities shortly before the UV exposure. In other words, salicylic acid was still on the surface of the skin during the sunlight exposure and was able to act as a sunscreen by absorbing the UV energy directly. You can not use salicylic acid as a sunscreen though - to get any practical sun protection, you would need to use a concentration way above 2% that is commonly used in cosmetics, and this concentration would cause too much exfoliation and irritation for frequent use.
The Effects of Topically Applied Glycolic Acid and Salicylic Acid on Ultraviolet Radiation-Induced Erythema, DNA Damage and Sunburn Cell Formation in Human Skin - https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2791365/
Photo by Kent Pilcher