The prime minister also indicated that EU leaders have shown an interest in giving the UK access to a key intelligence database used to identify people seeking asylum.
His comments, made at an Interpol annual general assembly in Glasgow, came days before he meets EU leaders including Giorgia Meloni of Italy to discuss future cooperation.
Emphasising his "personal mission to smash the people-smuggling gangs", Starmer said he plans to improve on previous negotiations with European partners.
"I don't think we did as well as we should have done when we came to the question of cross-border crime and security, and that's why we want to improve on it. And that can be done at the EU level.
"I'm very keen to get the UK into a leadership position when it comes to the actual operations, particularly in relation to smuggling and putting people into small boats across the Channel," he said.
It is understood that the government is seeking access to Eurodac, which stores the fingerprints of asylum seekers and migrants who have entered a European country.
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