This statement is partially valid. To assess the validity of the statement two aspects of the war should be taken in consideration. During the First World War modern, heavier and weapons more lethal than ever before were used for the first time. Traditional weapons were redesigned, making each soldier deadlier than before. However old tactics going back to the 19th century were used to combat the enemies.
The First World War is often referred to as "a 20th century war fought with 19th century tactics". It was the first "world" war in the true sense of the word, involving the whole of Europe and even, towards its end, the United States of America.
"Modern" it became indeed when one considers the new, deadlier weapons that were developed just before and during this war: Tanks, airplanes, machine guns, chemical weapons are the most widely known by the general public.
However, "old fashioned" tactics were used in every day combat tactics.
Trench warfare and massive offensives designed by old world general sipping cognac in their comfortable leather armchairs, sending foot soldiers to be butchered to an almost certain death as if they were young boys simulating a battle with lead soldiers, are there to remind us that this war was not "franche et joyeuse" (clearcut: "brave and joyous" is the best translation I could think of).
World War one is indeed the first modern war, as far as the weapons developed and used were concerned, but in my opinion the tactics used in the war were closer to those of the Napoleonic wars than the later 20th century wars.
A modern aspect of the First World War was the weaponry used or developed. During the First World War a wide range of new weapons appeared. Old weapons were rediscovered and redesigned...