From a security perspective, both providers are completely secure:
GoCardless works with over 3000 businesses, including The Guardian, the FT and Funding Circle. They are a BACS-approved bureau and are regulated by the Financial Conduct Authority as an Authorised Payment Institution. All monies collected by GoCardless are held in a designated client monies account. More information can be found on their website: https://gocardless.com/faq/merchants/security/
Stripe has been audited by a PCI-certified auditor and is certified to PCI Service Provider Level 1, which is the most stringent level of certification available. More information can be found on their website: https://stripe.com/docs/security/stripe