Baselli was born in Brescia and joined Atalanta's youth system in 2000, aged 8. In the 2010–11season he was promoted to the club's Primavera, for which he appeared in 22 league matches and scored five goals. He also collected some appearances on the bench for the first team both in Serie B and Coppa Italia.
On 13 July 2011 he was sold to Serie B club Cittadella in a co-ownership deal for a nominal fee of €500, for which he made his debut in a 1–2 loss in Coppa Italia to Fiorentina. He made his debut in the league on 27 August, starting in a 2–1 home success against AlbinoLeffe. He finished the 2011–12 season with 13 appearances in the league, including 10 as a starter. After the renewal of the co-ownership between the two clubs he remained in Veneto the following season, during which he took part in two matches of Coppa Italia and 37 in the league, even scoring his professional goal on 18 May 2013, netting the winner in a home win over Ascoli.
On 20 June 2013 he returned to Atalanta, after the Bergamo side bought the other half of the player's rights for €803,500, with Cittadella offering a reported €50,000 less. He signed a four-year deal with Atalanta, and made his debut for the club on 18 August, playing the last 17 minutes and assisting Giuseppe De Luca in the last goal of a 3–0 home win over Bari. On 1 September Baselli made his Serie A debut, coming on as a second-half substitute in a 2–0 home success over Torino; he was handed his first start in the main category of Italian football thirteen days later, in a 0–2 loss against Napoli. He scored his first goal in Serie A on 10 May 2015 in Palermo, scoring the 1–0 against the Rosaneri.
Torino
On 10 July2015, he was sold to Torino, in the same transfer that also brought Davide Zappacosta to Piedmont, for a total of €10.3 million. Baselli made his competitive debut for Torino on 17 August in the third round of Coppa Italia against Pescara, scoring Torino's first goal in a 4–1 win. On 23 August, he scored his first goal in Serie A with Torino in a 2–1 win away to Frosinone. He concluded the 2014–15 season with 35 appearances and four goals. On 20 August 2016, he scored his first goal of the 2016–17 season in a 3–2 loss against Milan at the San Siro. He concluded the season with 37 appearances and six goals. On 12 July 2017, he renewed his contract with Torino until 2022.
According to his agent Giuseppe Riso: "His role is certainly as a central midfielder, not in front of the defence. Over time he will be able to play as a metodista however right now his movement is quite dangerous when he makes runs into the opposing area. He is a 'box to box midfielder'". An elegant, technically gifted, and complete midfielder, Baselli is a clean tackler and a competent passer with both feet; due to his attributes, he is capable of winning balls, scoring goals, or creating chances for his team.
