Beck is the grandson of Ian Rush's sister, Carole. The Welsh left-back was snapped up from Stoke when he was 13 years of age. Beck made a short stop in Liverpool's Under-18 squad before being promoted to the under-23 side. Beck signed a new long-term contract in the summer of 2021 and made his debut as a late sub in the 2-0 win at Preston on 27 October 2021. He also came on in the following round against Leicester at Anfield and was on the bench for three Premier League games. He was initially loaned to Portuguese side Famalicao for 2022/23 but after failing to make it off the bench for the opening three games was recalled at the end of August and loaned to Bolton Wanderers. That spell was cut short by mutual consent in January after he failed to get regular game time.

Beck joined Dundee on loan for the 2023/24 season. However, he was recalled in January due to injuries to both Andy Robertson and Kostas Tsimikas. He had impressed during his spell in Scotland, scoring twice in seventeen league appearances. Beck made a late substitute appearance at Bournemouth on 21 January but with Robertson returning from injury before the end of the month, he rejoined Dundee on loan for the rest of the campaign. Beck's performances again raised eyebrows and he was named in the PFA team of the year. However, his loan spell was brought to a close in April when it was confirmed a groin injury would rule him out for the rest of the season. He was loaned to Blackburn for 2024/25 and his energetic displays down the left flank made him a firm favourite at Ewood Park. However, a hamstring injury sustained in February ruled him out for two months and he suffered a recurrence within just 13 minutes of his return to action, leading to his spell there coming to an early end in April. With Milos Kerkes arriving from Bournemouth in the summer of 2025, Beck's opportunities for the first team appeared limited and a season long loan at Derby County was agreed for 2024/25.