Generally yes. Big Fish Games is a legitimate, long-running (since 2002) casual game publisher. Their site and launcher use HTTPS/TLS 1.2+, PCI-DSS compliant payment processing (Stripe, Adyen), and a GDPR/CCPA-compliant privacy policy. No major breaches have been publicly disclosed as of 2024.
Treat Big Fish like any e-commerce site: standard caution keeps you safe.