She wanted them to work the issues out between themselves, she mistakenly believed that Meredith and Orsino could find a peaceful path through this conflict. She was unaware that Red Lyrium was making Meredith less sane than she used to be. Remember, Meredith actually rejected a proposal by Alrik to make all mages Tranquil, that implies that at one point the Knight-Commander was a reasonable authority figure.
I am amazed at the number of people that try to justify Anders actions by villifying the Grand Cleric espeically when Anders confession makes it clear he killed her because she could and had been a brake on Meredith. As long as the Grand Cleric was around some form of peace/compromise could be reached, he knew that and thus killed her to ensure that no peace would be found.
Meredith was the gasoline and Orsino was the fire. It was only time before the gas and the fire meet up and create total destruction. Elthina could've done something about it, but she did nothing. She did not treat the situation seriously, as she should of done. In fact she treated Orsino and Meredith more like children rather than a business acquaintances.
I tend to justify to what Anders did because Elthina was a monstrous criminal disgusted in a what appears to be an innocent old woman's body. The fact that she did not had Sister/Mother Petrice immediately executed for killing the Viscounts son and caused the Qunari to declare war against the people of Kirkwall that resulted in the deaths of many only tells me that the Chantry takes care of each other behind closed doors. If you side with Petrice the only punishment she received was a demotion of her title; from mother to sister.
After presenting evidence of Ser Alriks crimes of illegally making mages tranquil to Elthina she showed more sympathy for her criminal templar rather than to the fallen victims of Alriks. She then called you a murderer for giving Alrik some rough justice.
Anders wouldn't had blown the Chantry up if Elthina had better communication with the mages that was fair.