The Cantabrian coast is famously wet, and Gijón does not disappoint in that regard: it rains here in every month, and the sky is overcast for a significant proportion of the year. That said, the summer months of June through September are the best window — temperatures in the 18–24°C range, longer dry spells, and the full beach and festival season in operation. Semana Negra happens in July; the Gijón International Film Festival (FICX) runs in November, which is a different kind of reason to visit.
July and August are peak season: Spanish holidaymakers from Madrid and beyond arrive to escape the heat of the interior (Gijón's mild Atlantic temperatures are the point, not a disappointment), the beach is packed, and accommodation is at a premium. June and September offer the same green Asturian landscape and most of the same facilities with fewer crowds and lower prices. Winter is genuinely bleak weather-wise, but the city functions normally, the sidrerías are open, the museums are uncrowded, and the coastal storms give the beach and the Cimadevilla headland a completely different character from the summer version — one that the postcards don't show but that Gijón people themselves know.