The word Paul uses here, is αναξιως, a compound word consisting of αν + αξιως. So, it means "not" + "appropriately" or "inappropriately".
In 1 Corinthians 11:24-25, Paul records how Jesus instituted the memorial service. He:
gave thanks for the bread
broke the bread and invited the disciples to eat it in order to recall his body that was about to be broken for them (flesh torn away by the scourge, etc)
gave thanks for the wine ("in like manner")
drank the wine and invited the disciples to drink it in order to recall his blood that was about to be shed for them.
It is a solemn memorial service, so any attitude or action that didn't befit the solemnity of the occasion would be considered as behaving "unsatisfactorily", or "unworthily" or in an "unworthy manner".
To shed further light on this word, it is worth investigating Paul's use of αξιως in his letters.
In Romans 16:2, Paul asks in regard to Phoebe:
That ye receive her in the Lord, as becometh saints ...
In Ephesians 4:1, Paul exhorts the church to:
... walk worthy of the vocation wherewith ye are called.
In Philippians 1:17, concerning their conversation, Paul commends that it be exclusively:
... as it becometh the gospel of Christ.
In Colossians 1:9-10, Paul prays that the church be filled with the knowledge of God's will in all wisdom and spiritual understanding, so that they:
... might walk worthy of the Lord unto all pleasing.
In 1 Thessalonians 2:12, Paul exhorts those in the church that they:
... would walk worthy of God ...
In 2 Thessalonians 1:11, Paul prays that God count the church:
... worthy of this calling
So, αναξιως would describe any attitude or action that would:
not befit the saints -- those who have made a life-long commitment to adhere to God's expectations and Jesus' modeling of such.
impede the work that Jesus has called people to accomplish -- to be the salt and light of the world.
confound or obscure the Gospel of Christ -- the good news that the treasures of heaven are no longer exclusively available to a single people group, but are available to everyone.
project or promote disobedience.