Barca have topped the table in seven of the last 10 seasons but have found European success harder, reaching only one of the last seven Champions League finals.
Valverde's men won the double last term, adding Copa Del Rey glory to the La Liga title, but their 2017-18 campaign was marred by an extraordinary European exit.
Despite taking a 4-1 lead against Roma in the quarter-finals of the Champions League, the Catalan giants contrived to lose 3-0 in the Italian capital to crash out on away goals.
Rivals Real Madrid subsequently collected a third consecutive Champions League crown but Valverde insists he will not switch focus to European matters in 2018-19.
"We prioritise everything, we will always fight for the Champions League," Valverde said ahead of Saturday's La Liga opener against Deportivo Alaves.
"Barca will always be like that, we are very excited but the way to win the Champions League is to win the league. You focus on being better for a year, and if so, you are closer to winning the Champions League.
"It will be a match similar to last year's, they [Alaves] are a team that plays good games at the Camp Nou, the last team to win here in the league, and since Abelardo arrived, they have remained faithful to a style.
"I feel the responsibility at every moment, like last year we had it after having problems in August and living a difficult season, the challenges are repeated and the difficult thing is not to win, but to win again."
Valverde is unsure whether new signing Arturo Vidal will be fit enough to face Alaves, while defender Gerard Pique is another doubt for the champions.
"I do not know, if he's one 100 per cent, we'll have to wait for that," Valverde said of the Chile midfielder. "Because there will also be others that will be ready for 90 and will not play the 90.
"He has been inactive for four months and there are players with whom he has some competition. [Pique] will be training and we will see how he is doing."
Arthur is in line to make his league debut for Barcelona and the former Gremio midfielder was named in the Brazil squad for the first time on Friday, alongside Philippe Coutinho.
"For our players to go to the national team is good for them and also for us," Valverde added. "Philippe is very important for us and for Brazil, and in the case of Arthur it is a great joy, we hope he will help us a lot."